[ { "id": 1, "problem_number": "MPP-001", "title": "P versus NP Problem", "statement": "Does $P = NP$? More formally: if the solution to a problem can be quickly verified (in polynomial time), can the solution also be quickly found (in polynomial time)?", "background": "The P versus NP problem is a major unsolved problem in computer science. It asks whether every problem whose solution can be quickly verified can also be quickly solved. The Clay Mathematics Institute has offered a $1,000,000 prize for a correct solution.", "difficulty_level_id": 5, "status": "open", "proposed_by": "Stephen Cook", "proposed_year": 1971, "category_id": 15, "set_id": 1, "view_count": 1523, "favorite_count": 89, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2, "problem_number": "MPP-002", "title": "The Riemann Hypothesis", "statement": "Do all non-trivial zeros of the Riemann zeta function $\\zeta(s)$ have real part equal to $\\frac{1}{2}$?", "background": "The Riemann hypothesis, proposed by Bernhard Riemann in 1859, concerns the distribution of prime numbers. It states that all non-trivial zeros of the Riemann zeta function lie on the critical line $\\Re(s) = \\frac{1}{2}$. This is one of the most important open problems in mathematics, with profound implications for number theory.", "difficulty_level_id": 5, "status": "open", "proposed_by": "Bernhard Riemann", "proposed_year": 1859, "category_id": 1, "set_id": 1, "view_count": 2341, "favorite_count": 156, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 3, "problem_number": "MPP-003", "title": "Yang–Mills Existence and Mass Gap", "statement": "Prove that Yang–Mills theory exists and has a mass gap on $\\mathbb{R}^4$, meaning the quantum particles have positive masses.", "background": "This problem concerns quantum field theory and seeks to establish a rigorous mathematical foundation for Yang–Mills theories, which describe fundamental forces in particle physics. A solution would require proving the existence of these theories in four-dimensional spacetime and showing they predict a mass gap.", "difficulty_level_id": 5, "status": "open", "proposed_by": "Yang Chen-Ning and Robert Mills", "proposed_year": 1954, "category_id": 16, "set_id": 1, "view_count": 1234, "favorite_count": 78, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 4, "problem_number": "MPP-004", "title": "Navier–Stokes Existence and Smoothness", "statement": "Prove or give a counterexample: Do solutions to the Navier–Stokes equations in three dimensions always exist and remain smooth for all time?", "background": "The Navier–Stokes equations describe the motion of fluids. While solutions exist for short times and in two dimensions, the question of whether smooth solutions exist globally in three dimensions remains open. This has profound implications for understanding turbulence and fluid dynamics.", "difficulty_level_id": 5, "status": "open", "proposed_by": "Claude-Louis Navier and George Gabriel Stokes", "proposed_year": 1822, "category_id": 9, "set_id": 1, "view_count": 1456, "favorite_count": 89, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 5, "problem_number": "MPP-005", "title": "Birch and Swinnerton-Dyer Conjecture", "statement": "The conjecture relates the rank of the abelian group of rational points of an elliptic curve to the order of zero of the associated L-function at $s=1$.", "background": "This conjecture connects the arithmetic of elliptic curves (solutions to equations of the form $y^2 = x^3 + ax + b$) to the behavior of certain complex functions. It has deep connections to number theory and algebraic geometry.", "difficulty_level_id": 5, "status": "open", "proposed_by": "Bryan Birch and Peter Swinnerton-Dyer", "proposed_year": 1960, "category_id": 1, "set_id": 1, "view_count": 1123, "favorite_count": 67, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 6, "problem_number": "MPP-006", "title": "Hodge Conjecture", "statement": "On a projective non-singular algebraic variety over $\\mathbb{C}$, any Hodge class is a rational linear combination of classes of algebraic cycles.", "background": "The Hodge conjecture is a major open problem in algebraic geometry. It seeks to relate the topology of a smooth complex projective variety to its algebraic structure, specifically asserting that certain topological cycles are actually algebraic.", "difficulty_level_id": 5, "status": "open", "proposed_by": "William Vallance Douglas Hodge", "proposed_year": 1950, "category_id": 5, "set_id": 1, "view_count": 987, "favorite_count": 54, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 8, "problem_number": "NT-001", "title": "Odd Perfect Numbers", "statement": "Does there exist an odd perfect number? A perfect number is a positive integer that is equal to the sum of its proper divisors (excluding itself). For example, $6 = 1 + 2 + 3$ is perfect.", "background": "While many even perfect numbers are known (the first few are 6, 28, 496, 8128), no odd perfect number has ever been found, despite extensive computer searches. It has been proven that if one exists, it must be greater than $10^{1500}$ and have at least 101 prime factors.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 543, "favorite_count": 34, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 9, "problem_number": "NT-002", "title": "Collatz Conjecture", "statement": "Starting with any positive integer $n$, repeatedly apply the function: if $n$ is even, divide by 2; if $n$ is odd, multiply by 3 and add 1. Does this process always eventually reach 1?", "background": "Also known as the 3n+1 problem, this deceptively simple conjecture has been verified for all starting values up to $2^{68}$ but remains unproven. Paul Erdős said about it: \"Mathematics may not be ready for such problems.\"", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 892, "favorite_count": 67, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 11, "problem_number": "NT-003", "title": "Twin Prime Conjecture", "statement": "Are there infinitely many twin primes? Twin primes are pairs of primes that differ by 2, such as (3, 5), (5, 7), (11, 13), (17, 19), (29, 31).", "background": "The twin prime conjecture is one of the oldest unsolved problems in number theory. In 2013, Yitang Zhang proved that there are infinitely many pairs of primes that differ by at most 70 million. This bound has since been reduced to 246, but the gap of 2 remains unproven.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 1234, "favorite_count": 89, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 12, "problem_number": "NT-004", "title": "Goldbach's Conjecture", "statement": "Every even integer greater than 2 can be expressed as the sum of two primes.", "background": "Proposed by Christian Goldbach in 1742, this conjecture has been verified computationally for all even integers up to very large numbers. The weak Goldbach conjecture (every odd number greater than 5 is the sum of three primes) was proved by Harald Helfgott in 2013, but the strong version remains open.", "difficulty_level_id": 4, "status": "open", "proposed_by": "Christian Goldbach", "proposed_year": 1742, "category_id": 1, "view_count": 1567, "favorite_count": 112, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 13, "problem_number": "NT-005", "title": "ABC Conjecture", "statement": "For any $\\epsilon > 0$, there exist only finitely many triples $(a, b, c)$ of coprime positive integers with $a + b = c$ such that $c > \\text{rad}(abc)^{1+\\epsilon}$, where $\\text{rad}(n)$ is the product of distinct prime factors of $n$.", "background": "The ABC conjecture, formulated by Joseph Oesterlé and David Masser in 1985, has profound implications for number theory. Shinichi Mochizuki claimed a proof in 2012 using his \"inter-universal Teichmüller theory,\" but the proof remains controversial and not widely accepted.", "difficulty_level_id": 5, "status": "open", "proposed_by": "Joseph Oesterlé and David Masser", "proposed_year": 1985, "category_id": 1, "view_count": 876, "favorite_count": 45, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 10, "problem_number": "COMB-001", "title": "The Hadwiger-Nelson Problem", "statement": "What is the minimum number of colors needed to color the points of the plane such that no two points at distance 1 have the same color?", "background": "It is known that this chromatic number is between 5 and 7. In 2018, Aubrey de Grey proved it is at least 5, but whether it is 5, 6, or 7 remains unknown.", "difficulty_level_id": 3, "status": "open", "category_id": 2, "view_count": 421, "favorite_count": 28, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 14, "problem_number": "GT-001", "title": "Hadwiger Conjecture", "statement": "Every graph with chromatic number $k$ has a $K_k$ minor (where $K_k$ is the complete graph on $k$ vertices).", "background": "The Hadwiger conjecture, proposed in 1943, generalizes the four color theorem. It has been proved for $k \\leq 6$ but remains open for $k \\geq 7$. The case $k=5$ is equivalent to the four color theorem.", "difficulty_level_id": 4, "status": "open", "proposed_by": "Hugo Hadwiger", "proposed_year": 1943, "category_id": 3, "view_count": 654, "favorite_count": 38, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 15, "problem_number": "GT-002", "title": "Reconstruction Conjecture", "statement": "Every finite simple graph on at least 3 vertices is uniquely determined by its vertex-deleted subgraphs.", "background": "The reconstruction conjecture asks whether a graph can be uniquely reconstructed from the multiset of all its vertex-deleted subgraphs. Proposed by Stanisław Ulam in 1942, it has been verified for many classes of graphs but remains open in general.", "difficulty_level_id": 3, "status": "open", "proposed_by": "Stanisław Ulam", "proposed_year": 1942, "category_id": 3, "view_count": 432, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 18, "problem_number": "TOP-001", "title": "Smooth 4-Dimensional Poincaré Conjecture", "statement": "Is every smooth homotopy 4-sphere diffeomorphic to the standard 4-sphere $S^4$?", "background": "The smooth Poincaré conjecture in dimension 4 is the only remaining case of the generalized Poincaré conjecture. It has been solved in all other dimensions: dimension 3 by Perelman, higher dimensions by Smale, Freedman, and others. The 4-dimensional case is particularly difficult due to exotic smooth structures.", "difficulty_level_id": 5, "status": "open", "category_id": 7, "view_count": 789, "favorite_count": 42, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 19, "problem_number": "GEO-002", "title": "Sphere Packing in Higher Dimensions", "statement": "What is the densest packing of congruent spheres in $n$ dimensions for $n \\geq 4$?", "background": "The sphere packing problem asks for the densest arrangement of non-overlapping spheres. Maryna Viazovska solved it for dimension 8 in 2016, and she with collaborators solved it for dimension 24 in 2017. The problem remains open for most other dimensions.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 456, "favorite_count": 27, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 20, "problem_number": "ALG-001", "title": "Inverse Galois Problem", "statement": "Is every finite group the Galois group of some Galois extension of the rational numbers $\\mathbb{Q}$?", "background": "The inverse Galois problem asks whether every finite group can be realized as the Galois group of a polynomial equation with rational coefficients. It has been solved for many classes of groups, including all symmetric and alternating groups, but remains open in general.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 543, "favorite_count": 29, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 21, "problem_number": "ALG-002", "title": "Kaplansky's Conjectures", "statement": "A set of conjectures about group rings: (1) Zero divisor conjecture: If $G$ is a torsion-free group and $K$ is a field, then $K[G]$ has no zero divisors. (2) Idempotent conjecture: The only idempotents in $K[G]$ are 0 and 1. (3) Unit conjecture: The only units in $\\mathbb{Z}[G]$ are of the form $\\pm g$ for $g \\in G$.", "background": "These conjectures, proposed by Irving Kaplansky in the 1940s, concern the algebraic structure of group rings. They have been verified for many classes of groups but remain open in general. The zero divisor conjecture is related to the Atiyah conjecture.", "difficulty_level_id": 4, "status": "open", "proposed_by": "Irving Kaplansky", "proposed_year": 1940, "category_id": 4, "view_count": 321, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 22, "problem_number": "SET-001", "title": "Continuum Hypothesis", "statement": "There is no set whose cardinality is strictly between that of the integers and the real numbers.", "background": "The continuum hypothesis was the first of Hilbert's 23 problems. Kurt Gödel (1940) and Paul Cohen (1963) proved it is independent of ZFC set theory: it can neither be proved nor disproved from the standard axioms. Whether to accept it as an axiom remains a philosophical question.", "difficulty_level_id": 5, "status": "open", "proposed_by": "Georg Cantor", "proposed_year": 1878, "category_id": 10, "view_count": 1234, "favorite_count": 67, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 25, "problem_number": "NT-006", "title": "Legendre's Conjecture", "statement": "For every positive integer $n$, there exists a prime number between $n^2$ and $(n+1)^2$.", "background": "This conjecture about the distribution of prime numbers was proposed by Adrien-Marie Legendre in 1808. Despite significant progress in prime number theory, including the prime number theorem, this simple statement remains unproven.", "difficulty_level_id": 3, "status": "open", "proposed_by": "Adrien-Marie Legendre", "proposed_year": 1808, "category_id": 1, "view_count": 432, "favorite_count": 26, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 26, "problem_number": "NT-007", "title": "Are there infinitely many Mersenne primes?", "statement": "Are there infinitely many prime numbers of the form $M_p = 2^p - 1$ where $p$ is prime?", "background": "Mersenne primes are primes of the form $2^p - 1$. As of 2024, only 51 Mersenne primes are known, with the largest being $2^{82,589,933} - 1$. It is conjectured that infinitely many exist, but this remains unproven. They are important for computational number theory and the GIMPS distributed computing project.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 654, "favorite_count": 38, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 27, "problem_number": "NT-008", "title": "Are there infinitely many perfect powers in the Fibonacci sequence?", "statement": "Besides 1, 8, and 144, are there any other perfect powers (numbers of the form $a^b$ where $a, b > 1$) in the Fibonacci sequence?", "background": "The Fibonacci sequence has only three known perfect powers: $F_1 = F_2 = 1 = 1^n$, $F_6 = 8 = 2^3$, and $F_{12} = 144 = 12^2$. It is conjectured that these are the only ones, but this remains unproven despite extensive computational searches.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 345, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 28, "problem_number": "NT-009", "title": "Gilbreath's Conjecture", "statement": "Starting with the sequence of primes and repeatedly taking absolute differences of consecutive terms, the first term of each row is always 1.", "background": "Norman Gilbreath observed in 1958 that applying the forward difference operator to the sequence of primes appears to always yield 1 as the first element. Despite being verified computationally for the first $10^{13}$ primes, no proof exists.", "difficulty_level_id": 3, "status": "open", "proposed_by": "Norman Gilbreath", "proposed_year": 1958, "category_id": 1, "view_count": 287, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 29, "problem_number": "COMB-003", "title": "Ramsey Number R(5,5)", "statement": "What is the exact value of $R(5,5)$, the smallest number $n$ such that any 2-coloring of the edges of $K_n$ contains a monochromatic $K_5$?", "background": "Ramsey theory asks how large a structure must be to guarantee a certain property. The Ramsey number $R(5,5)$ is known to lie between 43 and 48, but the exact value remains unknown. As Joel Spencer said, \"Erdős asks us to imagine an alien force, demanding the value of $R(5,5)$ or they will destroy our planet... our best strategy is to get our best computers and mathematicians working on it.\"", "difficulty_level_id": 3, "status": "open", "category_id": 2, "view_count": 543, "favorite_count": 32, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 30, "problem_number": "COMB-004", "title": "The Lonely Runner Conjecture", "statement": "For any $n$ runners on a circular track with distinct constant speeds, each runner is \"lonely\" (distance at least $1/n$ from all others) at some time.", "background": "This combinatorial conjecture, proposed by J.M. Wills in 1967, has been verified for up to 7 runners but remains open for 8 or more. It has connections to Diophantine approximation and view-obstruction problems.", "difficulty_level_id": 3, "status": "open", "proposed_by": "J.M. Wills", "proposed_year": 1967, "category_id": 2, "view_count": 234, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 31, "problem_number": "GT-003", "title": "The Graceful Tree Conjecture", "statement": "Every tree can be gracefully labeled: vertices can be assigned distinct labels from $\\{0, 1, \\ldots, |E|\\}$ such that edge labels (absolute differences) are all distinct.", "background": "The graceful labeling conjecture, proposed by Alexander Rosa in 1967, asks whether every tree admits a graceful labeling. It has been verified for many classes of trees including paths, caterpillars, and trees with at most 35 vertices, but remains open in general.", "difficulty_level_id": 3, "status": "open", "proposed_by": "Alexander Rosa", "proposed_year": 1967, "category_id": 3, "view_count": 321, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 32, "problem_number": "GEO-003", "title": "The Kakeya Conjecture", "statement": "A Kakeya set (containing a unit line segment in every direction) in $\\mathbb{R}^n$ must have Hausdorff dimension $n$.", "background": "The Kakeya conjecture concerns the minimal \"size\" of sets containing line segments in all directions. It has deep connections to harmonic analysis and PDE. The conjecture is known in dimension 2 but remains open for $n \\geq 3$. It would have important implications for the restriction conjecture in Fourier analysis.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 432, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 33, "problem_number": "GEO-004", "title": "The Moving Sofa Problem", "statement": "What is the largest area of a shape that can be maneuvered through an L-shaped corridor of unit width?", "background": "This classic problem in geometric optimization asks for the largest \"sofa\" that can navigate a right-angled hallway. The best known lower bound is approximately 2.2195 (Gerver's sofa, 1992), and the upper bound is $2\\sqrt{2} \\approx 2.8284$. The exact answer remains unknown.", "difficulty_level_id": 3, "status": "open", "category_id": 6, "view_count": 567, "favorite_count": 41, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 34, "problem_number": "TOP-002", "title": "The Volume Conjecture", "statement": "For a hyperbolic knot $K$, the limit of normalized colored Jones polynomials equals the hyperbolic volume of the knot complement.", "background": "The volume conjecture, proposed by Rinat Kashaev in 1995 and generalized by Murakami and Murakami in 2001, connects quantum invariants of knots to their classical geometric properties. It relates quantum topology to hyperbolic geometry and has been verified for many knots but remains unproven in general.", "difficulty_level_id": 4, "status": "open", "proposed_by": "Rinat Kashaev", "proposed_year": 1995, "category_id": 7, "view_count": 298, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 36, "problem_number": "AG-001", "title": "The Standard Conjectures on Algebraic Cycles", "statement": "A collection of conjectures about algebraic cycles on smooth projective varieties, including Lefschetz standard conjecture and Künneth standard conjecture.", "background": "The standard conjectures, formulated by Alexander Grothendieck in the 1960s, concern the theory of algebraic cycles and their cohomology. They would have profound consequences for algebraic geometry, including the independence of Betti numbers from the choice of Weil cohomology theory. The Hodge conjecture would follow from the Lefschetz standard conjecture.", "difficulty_level_id": 5, "status": "open", "proposed_by": "Alexander Grothendieck", "proposed_year": 1965, "category_id": 5, "view_count": 432, "favorite_count": 23, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 37, "problem_number": "AG-002", "title": "The Abundance Conjecture", "statement": "For a minimal model $X$ of non-negative Kodaira dimension, the canonical divisor $K_X$ is semi-ample.", "background": "The abundance conjecture is a major open problem in birational algebraic geometry and the minimal model program. It predicts that canonical divisors on minimal models have good positivity properties. The conjecture is known in dimension 3 and in many special cases, but remains open in dimension 4 and higher.", "difficulty_level_id": 4, "status": "open", "category_id": 5, "view_count": 298, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 38, "problem_number": "ALG-003", "title": "The Köthe Conjecture", "statement": "A ring has no non-zero nil ideal (an ideal all of whose elements are nilpotent) if and only if it has no non-zero nil one-sided ideal.", "background": "The Köthe conjecture concerns the structure of rings with nilpotent elements. Proposed by Gottfried Köthe in 1930, it remains one of the oldest open problems in ring theory. Various special cases have been resolved, but the general conjecture remains open.", "difficulty_level_id": 3, "status": "open", "proposed_by": "Gottfried Köthe", "proposed_year": 1930, "category_id": 4, "view_count": 234, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 40, "problem_number": "PDE-001", "title": "The Regularity Problem for Euler Equations", "statement": "Do solutions to the 3D Euler equations for incompressible fluid flow remain smooth for all time, given smooth initial data?", "background": "The Euler equations describe the motion of inviscid (frictionless) fluids. While the Navier-Stokes equations include viscosity and are a Millennium Prize Problem, the regularity of Euler equations is also a major open question. Finite-time blowup would have profound implications for fluid dynamics.", "difficulty_level_id": 4, "status": "open", "category_id": 9, "view_count": 456, "favorite_count": 26, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 41, "problem_number": "SET-002", "title": "Singular Cardinals Hypothesis", "statement": "If $\\kappa$ is a singular strong limit cardinal, then $2^\\kappa = \\kappa^+$.", "background": "The singular cardinals hypothesis, formulated by Paul Erdős and András Hajnal, concerns the behavior of the power set operation on infinite cardinals. It sits between the generalized continuum hypothesis and ZFC. Its consistency and independence status remains a major open problem in set theory.", "difficulty_level_id": 4, "status": "open", "proposed_by": "Paul Erdős and András Hajnal", "category_id": 10, "view_count": 287, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 42, "problem_number": "SET-003", "title": "Whitehead Problem", "statement": "Is every abelian group $A$ such that $\\text{Ext}^1(A, \\mathbb{Z}) = 0$ a free abelian group?", "background": "The Whitehead problem, posed by J.H.C. Whitehead in 1950, asks about the structure of certain abelian groups. Shelah proved in 1973 that the problem is independent of ZFC: it is true under the constructible universe axiom (V=L) but can be false under other set-theoretic axioms.", "difficulty_level_id": 4, "status": "open", "proposed_by": "J.H.C. Whitehead", "proposed_year": 1950, "category_id": 10, "view_count": 198, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 43, "problem_number": "CS-001", "title": "The Unique Games Conjecture", "statement": "For certain constraint satisfaction problems (unique games), it is NP-hard to approximate the maximum fraction of satisfiable constraints beyond a certain threshold.", "background": "The Unique Games Conjecture, proposed by Subhash Khot in 2002, has become central to computational complexity theory. If true, it would imply optimal hardness results for many approximation problems. Khot was awarded the Nevanlinna Prize in 2014 for this work, despite the conjecture remaining unresolved.", "difficulty_level_id": 4, "status": "open", "proposed_by": "Subhash Khot", "proposed_year": 2002, "category_id": 15, "view_count": 543, "favorite_count": 32, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 44, "problem_number": "CS-002", "title": "The Polynomial Hirsch Conjecture", "statement": "The diameter of the graph of a $d$-dimensional polytope with $n$ facets is bounded by a polynomial in $d$ and $n$.", "background": "The original Hirsch conjecture (diameter at most $n - d$) was disproved in 2010 by Francisco Santos. The polynomial Hirsch conjecture is a weaker version that remains open and is important for understanding the complexity of the simplex algorithm for linear programming.", "difficulty_level_id": 3, "status": "open", "category_id": 15, "view_count": 321, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 45, "problem_number": "HIL-012", "title": "Hilbert's 12th Problem: Extension of Kronecker-Weber Theorem", "statement": "Extend the Kronecker-Weber theorem on abelian extensions of the rationals to any base number field.", "background": "Hilbert's 12th problem, posed in 1900, asks for an explicit construction of abelian extensions of number fields, generalizing the Kronecker-Weber theorem which states that every abelian extension of the rationals is contained in a cyclotomic field. Despite significant progress in class field theory, the problem of finding explicit generators remains largely open.", "difficulty_level_id": 5, "status": "open", "proposed_by": "David Hilbert", "proposed_year": 1900, "category_id": 1, "set_id": 2, "view_count": 345, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 46, "problem_number": "HIL-016", "title": "Hilbert's 16th Problem: Topology of Algebraic Curves and Limit Cycles", "statement": "Determine the maximum number and relative positions of limit cycles for polynomial vector fields of degree $n$, and investigate the topology of real algebraic curves and surfaces.", "background": "Posed by David Hilbert in 1900, this two-part problem concerns (1) the topology of real algebraic varieties and (2) the limit cycles of planar polynomial differential equations. While it was shown in 1991-1992 by Ilyashenko and Écalle that polynomial vector fields have finitely many limit cycles, the question of whether there exists a finite upper bound H(n) for degree n remains open for any n > 1.", "difficulty_level_id": 5, "status": "open", "proposed_by": "David Hilbert", "proposed_year": 1900, "category_id": 6, "set_id": 2, "view_count": 432, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 47, "problem_number": "LAN-004", "title": "Landau's Fourth Problem: Primes of the Form n² + 1", "statement": "Are there infinitely many primes of the form $n^2 + 1$?", "background": "One of Landau's four problems presented at the 1912 International Congress of Mathematicians, this asks whether there are infinitely many primes that are one more than a perfect square. Examples include 2, 5, 17, 37, 101, 197, 257, 401. Despite being simple to state, it has remained unsolved for over 110 years and is considered \"unattackable at the present state of mathematics.\"", "difficulty_level_id": 4, "status": "open", "proposed_by": "Edmund Landau", "proposed_year": 1912, "category_id": 1, "set_id": 6, "view_count": 398, "favorite_count": 22, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 48, "problem_number": "SMA-004", "title": "Smale's 4th Problem: Integer Zeros of Polynomials", "statement": "Find efficient algorithms for deciding whether a polynomial with integer coefficients has an integer root.", "background": "Part of Stephen Smale's 18 problems for the 21st century (1998), this problem asks for polynomial-time algorithms to determine if a polynomial equation has integer solutions. This is related to Hilbert's 10th problem, which was shown to be undecidable in general, but specific cases and algorithms with better complexity remain of interest.", "difficulty_level_id": 4, "status": "open", "proposed_by": "Stephen Smale", "proposed_year": 1998, "category_id": 15, "set_id": 5, "view_count": 287, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 49, "problem_number": "SMA-005", "title": "Smale's 5th Problem: Height Bounds for Diophantine Curves", "statement": "Find effective uniform bounds for the heights of rational points on algebraic curves.", "background": "From Smale's 1998 list, this problem addresses the challenge of bounding the size of integer solutions to algebraic equations. While Faltings proved that curves of genus > 1 have finitely many rational points, the question of effective bounds on their heights remains a major open problem in arithmetic geometry.", "difficulty_level_id": 4, "status": "open", "proposed_by": "Stephen Smale", "proposed_year": 1998, "category_id": 5, "set_id": 5, "view_count": 234, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 50, "problem_number": "SMA-006", "title": "Smale's 6th Problem: Finiteness of Central Configurations", "statement": "For the Newtonian $n$-body problem with positive masses, are there only finitely many central configurations (relative equilibria) for each $n$?", "background": "This problem from Smale's 1998 list concerns celestial mechanics and asks whether gravitating bodies can have only finitely many stable equilibrium configurations. The question is known to be true for n = 3 and n = 4, but remains open for n ≥ 5. It connects classical mechanics with algebraic geometry.", "difficulty_level_id": 4, "status": "open", "proposed_by": "Stephen Smale", "proposed_year": 1998, "category_id": 6, "set_id": 5, "view_count": 198, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 51, "problem_number": "SMA-007", "title": "Smale's 7th Problem: Distribution of Points on the 2-Sphere", "statement": "What is the optimal arrangement of $n$ points on the 2-sphere to minimize energy for various potential functions?", "background": "Smale's 7th problem (1998) asks for the configuration that minimizes various energy functionals for points on a sphere. This includes the Thomson problem (electrons on a sphere) and related optimization questions. Solutions are known for small n and highly symmetric cases, but the general problem remains open and connects to crystallography and coding theory.", "difficulty_level_id": 3, "status": "open", "proposed_by": "Stephen Smale", "proposed_year": 1998, "category_id": 6, "set_id": 5, "view_count": 267, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 52, "problem_number": "SMA-009", "title": "Smale's 9th Problem: Linear Programming in Polynomial Time", "statement": "Find a strongly polynomial algorithm for linear programming.", "background": "Smale's 9th problem (1998) asks whether there exists an algorithm for linear programming whose running time is polynomial in the number of constraints and variables, independent of the bit-size of the input. While linear programming is solvable in polynomial time, no strongly polynomial algorithm is known for the general case.", "difficulty_level_id": 4, "status": "open", "proposed_by": "Stephen Smale", "proposed_year": 1998, "category_id": 15, "set_id": 5, "view_count": 312, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 53, "problem_number": "SMA-010", "title": "Smale's 10th Problem: The Pugh Closing Lemma", "statement": "Is the $C^r$ closing lemma true for dynamical systems?", "background": "The closing lemma in dynamical systems theory asks whether, for a diffeomorphism with a nonwandering point, there is an arbitrarily small perturbation that makes that point periodic. Pugh proved a $C^1$ version in 1967, but the $C^r$ version for r ≥ 2 remains open. This is Smale's 10th problem from his 1998 list.", "difficulty_level_id": 4, "status": "open", "proposed_by": "Stephen Smale", "proposed_year": 1998, "category_id": 6, "set_id": 5, "view_count": 176, "favorite_count": 9, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 54, "problem_number": "SMA-016", "title": "The Jacobian Conjecture", "statement": "If $F: \\mathbb{C}^n \\to \\mathbb{C}^n$ is a polynomial map with constant non-zero Jacobian determinant, then $F$ is invertible.", "background": "The Jacobian conjecture, proposed in 1939 and featured as Smale's 16th problem (1998), asks whether polynomial maps with nowhere-vanishing Jacobian determinant are necessarily invertible. Despite its elementary statement, it has resisted numerous attempts at proof. The conjecture is known to be true in dimension 1 and for maps of degree at most 2, but remains open in general.", "difficulty_level_id": 4, "status": "open", "proposed_by": "Ott-Heinrich Keller", "proposed_year": 1939, "category_id": 4, "set_id": 5, "view_count": 298, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 55, "problem_number": "COMB-005", "title": "Frankl's Union-Closed Sets Conjecture", "statement": "For every finite union-closed family of sets (other than the empty family), there exists an element that belongs to at least half of the sets.", "background": "Proposed by Péter Frankl in 1979, this is one of the best-known open problems in combinatorics. A union-closed family is a collection of sets closed under taking unions. Despite its simple statement, the conjecture has attracted many attempted proofs. Recent progress (2022-2024) has shown lower bounds: some element must be in at least 1% of sets (Gilmer 2022), improved to 38% of sets (2024).", "difficulty_level_id": 3, "status": "open", "proposed_by": "Péter Frankl", "proposed_year": 1979, "category_id": 2, "view_count": 389, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 56, "problem_number": "GEO-005", "title": "Inscribed Square Problem (Toeplitz Conjecture)", "statement": "Does every simple closed curve in the plane contain all four vertices of some square?", "background": "The inscribed square problem, also called the square peg problem or Toeplitz conjecture, was posed by Otto Toeplitz in 1911. It asks whether every Jordan curve (simple closed curve) inscribes a square. The conjecture is known to be true for convex curves, piecewise smooth curves, and many special cases, but remains open in full generality as of 2026.", "difficulty_level_id": 4, "status": "open", "proposed_by": "Otto Toeplitz", "proposed_year": 1911, "category_id": 6, "view_count": 432, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 57, "problem_number": "NT-010", "title": "Brocard's Problem", "statement": "Find all integer solutions to $n! + 1 = m^2$.", "background": "Brocard's problem asks for all positive integers n such that n! + 1 is a perfect square. Only three solutions are known: (4, 5), (5, 11), and (7, 71), corresponding to 4! + 1 = 25, 5! + 1 = 121, and 7! + 1 = 5041. It has been verified computationally that no other solutions exist for n < 10^9, but it remains unproven whether these are the only solutions.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 345, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 59, "problem_number": "GT-004", "title": "The Cycle Double Cover Conjecture", "statement": "Every bridgeless graph has a cycle double cover: a collection of cycles that covers each edge exactly twice.", "background": "The cycle double cover conjecture, proposed independently by Paul Seymour and Gábor Szekeres in the 1970s, is a major open problem in graph theory. It has been verified for many classes of graphs, including planar graphs and graphs with small genus. The conjecture is related to the snark conjecture and has connections to topology and algebraic graph theory.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 298, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 60, "problem_number": "NT-012", "title": "The Erdős-Straus Conjecture", "statement": "For every integer $n \\geq 2$, the equation $\\frac{4}{n} = \\frac{1}{x} + \\frac{1}{y} + \\frac{1}{z}$ has a solution in positive integers x, y, z.", "background": "The Erdős-Straus conjecture concerns Egyptian fractions (sums of unit fractions). Paul Erdős and Ernst G. Straus conjectured in 1948 that 4/n can always be expressed as the sum of three unit fractions. The conjecture has been verified for all n up to 10^17 and is known to hold for various infinite families, but a general proof remains elusive.", "difficulty_level_id": 3, "status": "open", "proposed_by": "Paul Erdős and Ernst G. Straus", "proposed_year": 1948, "category_id": 1, "view_count": 367, "favorite_count": 20, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 61, "problem_number": "HIL-006", "title": "Hilbert's 6th Problem: Axiomatization of Physics", "statement": "Develop a mathematical framework that axiomatizes physics, particularly mechanics, thermodynamics, and probability theory.", "background": "Hilbert's 6th problem (1900) calls for treating physics with the same mathematical rigor as geometry. While progress has been made (quantum mechanics axiomatization by von Neumann, some progress in quantum field theory), a complete axiomatization remains elusive, especially for areas like thermodynamics and a unified \"theory of everything.\"", "difficulty_level_id": 5, "status": "open", "proposed_by": "David Hilbert", "proposed_year": 1900, "category_id": 16, "set_id": 2, "view_count": 345, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 62, "problem_number": "HIL-013", "title": "Hilbert's 13th Problem: Seventh Degree Equations", "statement": "Prove that the general equation of the seventh degree cannot be solved using functions of only two variables.", "background": "Hilbert's 13th problem (1900) asks whether seventh-degree equations can be solved using continuous functions of two variables. Vladimir Arnold and Andrey Kolmogorov showed in 1957 that any continuous function can be represented using functions of two variables, which contradicts Hilbert's expectation. However, the problem of whether algebraic solutions exist with restrictions remains open.", "difficulty_level_id": 4, "status": "open", "proposed_by": "David Hilbert", "proposed_year": 1900, "category_id": 4, "set_id": 2, "view_count": 287, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 64, "problem_number": "SMA-012", "title": "Smale's 12th Problem: Centralizers of Diffeomorphisms", "statement": "Determine the structure of centralizers of generic diffeomorphisms.", "background": "Smale's 12th problem (1998) concerns the algebraic structure of diffeomorphisms that commute with a given diffeomorphism. The centralizer of a dynamical system reveals its symmetries. Smale conjectured that for generic diffeomorphisms, the centralizer should be trivial or nearly trivial.", "difficulty_level_id": 4, "status": "open", "proposed_by": "Stephen Smale", "proposed_year": 1998, "category_id": 6, "set_id": 5, "view_count": 176, "favorite_count": 9, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 66, "problem_number": "DARPA-002", "title": "The Dynamics of Networks", "statement": "Develop high-dimensional mathematics to model and predict behavior in large-scale distributed networks.", "background": "DARPA challenge 2 (2007) addresses the need for mathematical tools to understand massive networks like the internet, social networks, and biological networks. Traditional graph theory becomes inadequate at scale, requiring new mathematical frameworks for network dynamics.", "difficulty_level_id": 4, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 3, "set_id": 4, "view_count": 389, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 68, "problem_number": "DARPA-004", "title": "21st Century Fluids", "statement": "Extend classical fluid dynamics to handle complex substances like foams, suspensions, gels, and liquid crystals.", "background": "DARPA challenge 4 (2007) recognizes that most real-world fluids don't behave like the classical fluids of Navier-Stokes equations. New mathematics is needed for complex fluids with microstructure, non-Newtonian behavior, and multiphase dynamics.", "difficulty_level_id": 4, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 9, "set_id": 4, "view_count": 345, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 69, "problem_number": "DARPA-005", "title": "Biological Quantum Field Theory", "statement": "Apply quantum and statistical field theory methods to model and potentially control pathogen evolution.", "background": "DARPA challenge 5 (2007) proposes using the mathematical machinery of quantum field theory—developed for particle physics—to understand biological evolution and epidemiology. This could provide new ways to predict and control disease evolution.", "difficulty_level_id": 5, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 16, "set_id": 4, "view_count": 267, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 70, "problem_number": "DARPA-008", "title": "Beyond Convex Optimization", "statement": "Determine whether algebraic geometry can systematically replace linear algebra in optimization.", "background": "DARPA challenge 8 (2007) asks whether the powerful tools of algebraic geometry can extend optimization beyond the convex case. Most practical optimization problems are non-convex, and algebraic geometry may provide the framework for solving them systematically.", "difficulty_level_id": 4, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 15, "set_id": 4, "view_count": 312, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 71, "problem_number": "DARPA-012", "title": "Mathematics of Quantum Computing", "statement": "Develop the mathematics required to control the quantum world for computation.", "background": "DARPA challenge 12 (2007) calls for mathematical foundations of quantum computing, including quantum algorithms, quantum entanglement, and quantum error correction. While quantum computers exist, the mathematical theory of what they can compute and how to program them remains underdeveloped.", "difficulty_level_id": 5, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 15, "set_id": 4, "view_count": 543, "favorite_count": 32, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 72, "problem_number": "DARPA-013", "title": "Game Theory at Scale", "statement": "Create scalable mathematics for differential games, replacing traditional PDE approaches.", "background": "DARPA challenge 13 (2007) addresses the limitations of classical game theory and differential games when dealing with many players. New mathematical frameworks are needed for multi-agent systems, from autonomous vehicles to economic markets to military strategy.", "difficulty_level_id": 4, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 15, "set_id": 4, "view_count": 289, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 74, "problem_number": "DARPA-020", "title": "Computation at Scale", "statement": "Develop asymptotics for systems with massive degrees of freedom.", "background": "DARPA challenge 20 (2007) addresses the mathematical challenges of understanding systems with enormous numbers of variables—from climate models to protein folding to materials science. Traditional approaches fail at extreme scales, requiring new asymptotic methods.", "difficulty_level_id": 4, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 15, "set_id": 4, "view_count": 276, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 75, "problem_number": "DARPA-023", "title": "Fundamental Laws of Biology", "statement": "Identify governing principles for biological systems, analogous to physical laws.", "background": "DARPA challenge 23 (2007) poses perhaps the deepest question: Do fundamental mathematical laws govern biology the way physics is governed by laws? This challenge requires solutions to multiple preceding challenges and asks whether biology can be made as mathematically rigorous as physics.", "difficulty_level_id": 5, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 16, "set_id": 4, "view_count": 498, "favorite_count": 29, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 76, "problem_number": "DARPA-006", "title": "Computational Duality", "statement": "Use mathematical duality and geometry as foundations for developing novel computational algorithms.", "background": "DARPA challenge 6 (2007) explores whether duality principles from mathematics can lead to breakthrough algorithms. Dualities connect seemingly different mathematical structures and may reveal hidden computational efficiencies.", "difficulty_level_id": 4, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 15, "set_id": 4, "view_count": 198, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 77, "problem_number": "DARPA-007", "title": "Occam's Razor in Many Dimensions", "statement": "Find lower bounds for sensing complexity as data collection grows, addressing entropy maximization.", "background": "DARPA challenge 7 (2007) asks for mathematical principles governing data compression and sensing in high dimensions. As sensors become ubiquitous, we need mathematical theory for how much data is truly necessary.", "difficulty_level_id": 4, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 15, "set_id": 4, "view_count": 234, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 78, "problem_number": "DARPA-009", "title": "Physical Consequences of Perelman's Proof", "statement": "Apply Perelman's proof of the Poincaré conjecture to materials fabrication across scales.", "background": "DARPA challenge 9 (2007) asks how Grisha Perelman's breakthrough in understanding 3-dimensional geometry can inform materials science, from nanostructures to macro-scale fabrication.", "difficulty_level_id": 4, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 7, "set_id": 4, "view_count": 267, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 79, "problem_number": "DARPA-010", "title": "Algorithmic Origami and Biology", "statement": "Strengthen mathematical theory for isometric and rigid embedding relevant to protein folding.", "background": "DARPA challenge 10 (2007) connects origami mathematics to biology. Protein folding is like origami at molecular scales, and better mathematical theory could revolutionize drug design and protein engineering.", "difficulty_level_id": 4, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 6, "set_id": 4, "view_count": 298, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 80, "problem_number": "DARPA-011", "title": "Optimal Nanostructures", "statement": "Develop mathematics for creating optimal symmetric structures through nanoscale self-assembly.", "background": "DARPA challenge 11 (2007) seeks mathematical principles for designing nanostructures that self-assemble optimally. This combines crystallography, optimization, and molecular dynamics.", "difficulty_level_id": 4, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 6, "set_id": 4, "view_count": 223, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 81, "problem_number": "DARPA-015", "title": "The Geometry of Genome Space", "statement": "Establish appropriate distance metrics on genome space incorporating biological utility.", "background": "DARPA challenge 15 (2007) asks for a mathematical geometry of genetics. How \"far apart\" are two genomes? The answer depends on biology, not just counting mutations, requiring new geometric frameworks.", "difficulty_level_id": 4, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 6, "set_id": 4, "view_count": 245, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 82, "problem_number": "DARPA-016", "title": "Symmetries and Action Principles for Biology", "statement": "Extend understanding of symmetries and action principles in biology to include robustness, modularity, evolvability, and variability.", "background": "DARPA challenge 16 (2007) seeks to identify fundamental symmetry principles in biology analogous to those in physics. Why are biological systems robust yet evolvable? Are there variational principles governing life?", "difficulty_level_id": 5, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 16, "set_id": 4, "view_count": 312, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 83, "problem_number": "DARPA-017", "title": "Geometric Langlands and Quantum Physics", "statement": "Connect the Langlands program to fundamental physics symmetries.", "background": "DARPA challenge 17 (2007) explores deep connections between number theory (Langlands program) and quantum field theory. This could unify disparate areas of mathematics and physics.", "difficulty_level_id": 5, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 16, "set_id": 4, "view_count": 356, "favorite_count": 20, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 84, "problem_number": "DARPA-018", "title": "Arithmetic Langlands, Topology, and Geometry", "statement": "Explore homotopy theory's role in Langlands programs.", "background": "DARPA challenge 18 (2007) connects topology (homotopy theory) with the Langlands program in number theory. These connections could revolutionize both fields.", "difficulty_level_id": 5, "status": "open", "proposed_by": "DARPA", "proposed_year": 2007, "category_id": 7, "set_id": 4, "view_count": 289, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 85, "problem_number": "HIL-007", "title": "Hilbert's 7th Problem: Transcendence of Certain Numbers", "statement": "If $\\alpha$ is algebraic and irrational, and $\\beta$ is algebraic and irrational, is $\\alpha^\\beta$ transcendental?", "background": "Hilbert's 7th problem (1900) was largely solved by Gelfond and Schneider independently in 1934 (Gelfond-Schneider theorem). However, cases involving non-algebraic irrational exponents remain open. For example, whether $e^e$ or $\\pi^\\pi$ are transcendental is unknown.", "difficulty_level_id": 4, "status": "open", "proposed_by": "David Hilbert", "proposed_year": 1900, "category_id": 1, "set_id": 2, "view_count": 321, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 86, "problem_number": "HIL-009", "title": "Hilbert's 9th Problem: Reciprocity Laws", "statement": "Generalize the reciprocity law of number theory to arbitrary number fields.", "background": "Hilbert's 9th problem (1900) asks for extensions of quadratic reciprocity to general number fields. Emil Artin made progress with Artin reciprocity law (1927), but complete understanding of reciprocity in all cases remains an active research area.", "difficulty_level_id": 5, "status": "open", "proposed_by": "David Hilbert", "proposed_year": 1900, "category_id": 1, "set_id": 2, "view_count": 234, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 87, "problem_number": "HIL-011", "title": "Hilbert's 11th Problem: Quadratic Forms over Algebraic Number Fields", "statement": "Extend the theory of quadratic forms with algebraic numerical coefficients.", "background": "Hilbert's 11th problem (1900) concerns arithmetic of quadratic forms over number fields. Partial progress has been made through class field theory and the Hasse-Minkowski theorem, but general questions about representations remain open.", "difficulty_level_id": 4, "status": "open", "proposed_by": "David Hilbert", "proposed_year": 1900, "category_id": 1, "set_id": 2, "view_count": 198, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 88, "problem_number": "HIL-014", "title": "Hilbert's 14th Problem: Finite Generation of Rings", "statement": "Is the ring of invariants of a linear algebraic group acting on a polynomial ring always finitely generated?", "background": "Hilbert's 14th problem (1900) was answered negatively by Nagata in 1958, who found counterexamples. However, the problem remains interesting for special cases, and understanding when finite generation holds is an active area.", "difficulty_level_id": 4, "status": "open", "proposed_by": "David Hilbert", "proposed_year": 1900, "category_id": 4, "set_id": 2, "view_count": 176, "favorite_count": 9, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 89, "problem_number": "HIL-015", "title": "Hilbert's 15th Problem: Schubert's Enumerative Calculus", "statement": "Rigorously justify Schubert's enumerative geometry.", "background": "Hilbert's 15th problem (1900) calls for making Schubert's 19th century enumerative geometry rigorous. While intersection theory and Schubert calculus have been developed (Chow rings, Gromov-Witten theory), some classical problems remain open and new questions arise.", "difficulty_level_id": 4, "status": "open", "proposed_by": "David Hilbert", "proposed_year": 1900, "category_id": 5, "set_id": 2, "view_count": 267, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 90, "problem_number": "HIL-017", "title": "Hilbert's 17th Problem: Expression of Definite Forms", "statement": "Can every non-negative rational function be expressed as a sum of squares of rational functions?", "background": "Hilbert's 17th problem (1900) was solved affirmatively by Artin in 1927: every non-negative polynomial can be written as a sum of squares of rational functions. However, questions about minimal representations and related problems in real algebraic geometry remain active.", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Hilbert", "proposed_year": 1900, "category_id": 4, "set_id": 2, "view_count": 198, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 91, "problem_number": "HIL-018", "title": "Hilbert's 18th Problem: Polyhedra and Space-Filling", "statement": "Are there only finitely many essentially different space-filling convex polyhedra? Is there a polyhedron which tiles space but not in a lattice arrangement?", "background": "Hilbert's 18th problem (1900) has multiple parts. Non-lattice tilings (aperiodic tilings) were discovered by Heesch and others. The Kepler conjecture about sphere packing was proved by Hales. However, classification questions about space-filling polyhedra remain open.", "difficulty_level_id": 3, "status": "open", "proposed_by": "David Hilbert", "proposed_year": 1900, "category_id": 6, "set_id": 2, "view_count": 289, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 92, "problem_number": "GREEN-001", "title": "Large Sum-Free Sets", "statement": "Let $A$ be a set of $n$ positive integers. Does $A$ contain a sum-free set of size at least $n/3 + \\Omega(n)$, where $\\Omega(n) \\to \\infty$ as $n \\to \\infty$?", "background": "This is a pretty old and increasingly notorious problem, first mentioned over 50 years ago. The best known bounds are in Bourgain's paper, where he shows that there is necessarily a sum-free set of size at least $(n+2)/3$. In fact, Eberhard, Manners and Green (unpublished) worked out a proof that Problem 1 has a positive solution under certain structural assumptions. It is known that there do exist sets with no sum-free set of size larger than $(1/3 + o(1))n$. However, the $o(1)$ term in these results is more-or-less ineffective; it would be interesting to get a reasonable bound. [1] P. Erdős. Extremal problems in number theory, In Proc. Sympos. Pure Math., Vol. VIII, pages 181–189. Amer. Math. Soc., Providence, R.I., 1965. [2] J. Bourgain, Estimates related to sumfree subsets of sets of integers, Israel J. Math. 97 (1997), 71–92. [3] S. Eberhard, Følner sequences and sum-free sets, Bull. Lond. Math. Soc. 47 (2015), no. 1, 21–28. [4] S. Eberhard, B. J. Green and F. Manners, Sets of integers with no large sum-free subset, Ann. of Math. (2) 180 (2014), no. 2, 621–652.", "difficulty_level_id": 2, "status": "open", "proposed_by": "Erdős and Cameron", "category_id": 2, "set_id": 3, "view_count": 145, "favorite_count": 8, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 93, "problem_number": "GREEN-002", "title": "Restricted Sumset Problem", "statement": "Let $A \\subset \\mathbb{Z}$ be a set of $n$ integers. Is there a subset $S \\subset A$ of size $(\\log n)^{100}$ such that $S \\hat{+} S$ is disjoint from $A$?", "background": "Here $S \\hat{+} S$ denotes the restricted sumset $\\{s_1 + s_2 : s_1, s_2 \\in S, s_1 \\neq s_2\\}$. Problems of this type are also at least 50 years old, being once again mentioned (and attributed to joint discussions of Erdős and Moser). It is known from very recent work of Sanders that there is always such an $S$ with $|S| \\geq (\\log n)^{1+c}$. By contrast the best-known upper bound is due to Ruzsa, showing that one cannot in general hope to take $|S|$ bigger than $e^{C\\sqrt{\\log n}}$. [1] P. Erdős. Extremal problems in number theory, In Proc. Sympos. Pure Math., Vol. VIII, pages 181–189. Amer. Math. Soc., Providence, R.I., 1965. [2] T. Sanders, The Erdős-Moser sum-free set problem, Canad. J. Math. 73 (2021), no. 1, 63–107. [3] I. Z. Ruzsa, Sum-avoiding subsets. Ramanujan J., 9 (2005) (1-2):77–82.", "difficulty_level_id": 2, "status": "open", "proposed_by": "Erdős and Moser", "category_id": 2, "set_id": 3, "view_count": 123, "favorite_count": 7, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 95, "problem_number": "GREEN-005", "title": "Product-Free Sets in Finite Groups", "statement": "Which finite groups have the smallest largest product-free sets?", "background": "Kedlaya (2003) showed that every finite group $G$ of order $n$ has a product-free subset of size $\\gg n^{11/14}$, using the classification of finite simple groups. Understanding which groups achieve the minimum and improving bounds remains an open question connecting group theory and combinatorics.", "difficulty_level_id": 2, "status": "open", "proposed_by": "Kedlaya", "proposed_year": 2003, "category_id": 4, "set_id": 3, "view_count": 134, "favorite_count": 7, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 96, "problem_number": "GREEN-007", "title": "Ulam's Sequence", "statement": "Define Ulam's sequence $1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, \\ldots$ where $u_1 = 1, u_2 = 2$, and $u_{n+1}$ is the smallest number uniquely expressible as $u_i + u_j$ for $i < j \\leq n$. Does this sequence have positive density? Can one explain its curious Fourier properties?", "background": "Ulam's sequence exhibits mysterious quasi-periodic behavior in its Fourier transform. While it appears to have density around $0.07$, proving it has positive density remains open. The sequence's additive structure and apparent regularity in numerical experiments are not well understood theoretically.", "difficulty_level_id": 1, "status": "open", "proposed_by": "Stanisław Ulam", "category_id": 1, "set_id": 3, "view_count": 187, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 97, "problem_number": "GREEN-008", "title": "Almost Sum-Free Sets", "statement": "Suppose that $A \\subset [N]$ has no more than $\\varepsilon N^2$ solutions to $x + y = z$. Can one remove $\\varepsilon' N$ elements to leave a sum-free set, where $\\varepsilon' \\to 0$ as $\\varepsilon \\to 0$, with a reasonable bound?", "background": "It is known that one can remove $\\varepsilon' N$ elements to obtain a sum-free set, but the quantitative dependence of $\\varepsilon'$ on $\\varepsilon$ is very poor. Finding explicit reasonable bounds would significantly improve our understanding of the structure of almost sum-free sets.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 109, "favorite_count": 6, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 98, "problem_number": "GREEN-006", "title": "Sum-Free Subsets of [N]^d", "statement": "Fix an integer $d$. What is the largest sum-free subset of $[N]^d$?", "background": "This multi-dimensional generalization asks for the maximum size of a set in the $d$-dimensional grid with no solutions to $x + y = z$. Lepsveridze and Sun (2023) determined the constants $c_3, c_4, c_5$ and confirmed that the \"slice example\" is asymptotically optimal in these cases.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 118, "favorite_count": 7, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 99, "problem_number": "GREEN-009", "title": "Progressions in Subsets of Z/NZ", "statement": "Is $r_5(N) \\ll N(\\log N)^{-c}$? Is $r_4(\\mathbb{F}_5^n) \\ll N^{1-c}$ where $N = 5^n$?", "background": "Here $r_k(N)$ denotes the maximum size of a subset of $\\{1, \\ldots, N\\}$ with no $k$-term arithmetic progression. Kelley-Meka (2024) resolved the $k=3$ case. For $k \\geq 5$, Leng-Sah-Sawhney (2024) proved bounds of shape $r_k(N) \\ll Ne^{-(\\log \\log N)^{c_k}}$. Finding polynomial savings remains a central challenge in additive combinatorics.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 142, "favorite_count": 8, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 100, "problem_number": "GREEN-010", "title": "Roth's Theorem with Random Common Differences", "statement": "Let $S \\subset \\mathbb{N}$ be random. Under what conditions is Roth's theorem for progressions of length 3 true with common differences in $S$?", "background": "This asks when Roth's theorem holds if we restrict common differences to a random set. Briët and Castro-Silva (2023) advanced bounds for odd $k$. The problem explores how randomness interacts with additive structure.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 126, "favorite_count": 7, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 102, "problem_number": "GREEN-012", "title": "Tuples in Dense Sets", "statement": "Let $G$ be an abelian group of size $N$, and suppose that $A \\subset G$ has density $\\alpha$. Are there at least $\\alpha^{15}N^{10}$ tuples $(x_1, \\ldots, x_5, y_1, \\ldots, y_5) \\in G^{10}$ such that $x_i + y_j \\in A$ whenever $j \\in \\{i, i+1, i+2\\}$?", "background": "This problem asks about higher-order additive structures in dense sets. Deng-Tidor-Zhao (2023) considered this problem and conjectured a negative answer, suggesting the exponent might not be optimal.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 108, "favorite_count": 6, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 103, "problem_number": "GREEN-013", "title": "4-term APs in Fourier Uniform Sets", "statement": "Suppose that $A \\subset \\mathbb{Z}/N\\mathbb{Z}$ has density $\\alpha$ and is Fourier uniform (all Fourier coefficients of $1_A - \\alpha$ are $o(N)$). Does $A$ contain at least $\\gg \\alpha^{100}N^2$ 4-term arithmetic progressions?", "background": "Fourier uniformity means the set \"looks random\" from a Fourier perspective. The question asks if this forces many 4-APs. Deng-Tidor-Zhao (2023) conjectured a negative answer, suggesting Fourier uniformity alone may not suffice.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 115, "favorite_count": 7, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 104, "problem_number": "GREEN-015", "title": "Lipschitz AP-Free Graphs", "statement": "Does there exist a Lipschitz function $f : \\mathbb{N} \\to \\mathbb{Z}$ whose graph $\\Gamma = \\{(n, f(n)) : n \\in \\mathbb{Z}\\} \\subset \\mathbb{Z}^2$ is free of 3-term progressions?", "background": "This asks whether a \"smooth\" (Lipschitz) function can have a graph avoiding arithmetic progressions. The Lipschitz condition prevents wildly oscillating behavior, making AP-avoidance more constrained.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 121, "favorite_count": 7, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 105, "problem_number": "GREEN-016", "title": "Linear Equation x + 3y = 2z + 2w", "statement": "What is the largest subset of $[N]$ with no solution to $x + 3y = 2z + 2w$ in distinct integers $x, y, z, w$?", "background": "This asks about sets avoiding a specific linear configuration. Understanding which linear equations are easier or harder to avoid is a fundamental question in additive combinatorics.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 98, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 106, "problem_number": "GREEN-017", "title": "Progressions in F_3^n with Boolean Common Differences", "statement": "Suppose that $A \\subset \\mathbb{F}_3^n$ is a set of density $\\alpha$. Under what conditions on $\\alpha$ is $A$ guaranteed to contain a 3-term progression with nonzero common difference in $\\{0, 1\\}^n$?", "background": "This constrains the progression to have Boolean-like common differences. Bhangale-Khot-Minzer (2023) showed sets avoiding such progressions have density $\\ll_p (\\log \\log \\log n)^{-c_p}$, using extraordinarily difficult techniques.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 104, "favorite_count": 6, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 107, "problem_number": "GREEN-018", "title": "Corner Problem in Product Sets", "statement": "Suppose $G$ is a finite group, and let $A \\subset G \\times G$ be a subset of density $\\alpha$. Are there $\\gg_\\alpha |G|^3$ triples $x, y, g$ such that $(x, y), (gx, y), (x, gy)$ all lie in $A$?", "background": "This is a \"corner-type\" problem in the group product setting. Dense sets should contain many axis-aligned corners. The problem connects additive combinatorics with group theory.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 110, "favorite_count": 6, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 108, "problem_number": "GREEN-020", "title": "Multidimensional Szemerédi Theorem Bounds", "statement": "Find reasonable bounds for instances of the multidimensional Szemerédi theorem.", "background": "Szemerédi's theorem extends to multiple dimensions (finding combinatorial lines in dense sets). Pohoata-Zakharov (2024) improved bounds for skew corners to $N^{5/4}$. Quantitative bounds remain a major challenge.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 127, "favorite_count": 7, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 109, "problem_number": "GREEN-021", "title": "Large Sieve and Quadratic Sets", "statement": "Suppose that a large sieve process leaves a set of quadratic size. Is that set quadratic?", "background": "Sieve methods remove arithmetic structure from sets. This problem asks whether a set that \"survives\" a large sieve and has size $\\sim N^2$ must actually be a quadratic sequence or similar structured set. Understanding the structure of sieved sets is fundamental in analytic number theory.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 1, "view_count": 87, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 110, "problem_number": "GREEN-022", "title": "Small Sieve Maximal Sets", "statement": "Suppose that a small sieve process leaves a set of maximal size. What is the structure of that set?", "background": "When a small sieve (sieving by small primes) leaves the maximum possible density of survivors, what structure must the original set have? This connects sieve theory with the structural theory of sets in number theory.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 1, "view_count": 82, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 111, "problem_number": "GREEN-023", "title": "Large Cosets in Iterated Sumsets", "statement": "Suppose that $A \\subset \\mathbb{F}_2^n$ has density $\\alpha$. Does $10A$ contain a coset of some subspace of dimension at least $n - O(\\log(1/\\alpha))$?", "background": "This asks how many times we must add a set to itself before it contains a large subspace coset. Kosciuszko (2024), building on Konyagin, showed that $mA - mA$ contains a subspace of dimension $\\geq n - O(\\log^{3+\\eta}(1/\\alpha))$ for suitable $m$. The problem asks if fewer iterations suffice.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 93, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 112, "problem_number": "GREEN-024", "title": "Largest Coset in 2A", "statement": "Suppose that $A \\subset \\mathbb{F}_2^n$ has density $\\alpha$. What is the largest size of coset guaranteed to be contained in $2A$?", "background": "This asks for the largest affine subspace (coset) contained in the doubling $2A = A + A$. Unlike the previous problem about many iterations, this focuses on just $2A$. Determining the optimal bound is a fundamental question in additive combinatorics over $\\mathbb{F}_2^n$.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 88, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 113, "problem_number": "GREEN-025", "title": "Additive Complements and Cosets", "statement": "Suppose that $A \\subset \\mathbb{F}_2^n$ has an additive complement of size $K$. Does $2A$ contain a coset of codimension $O_K(1)$?", "background": "If $A$ has a small additive complement (a set $B$ with $A + B = \\mathbb{F}_2^n$), does this force $2A$ to contain a large coset? This problem explores the relationship between additive complements and the structure of sumsets.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 91, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 114, "problem_number": "GREEN-026", "title": "Partitions and Large Cosets", "statement": "Suppose that $\\mathbb{F}_2^n$ is partitioned into sets $A_1, \\dots, A_K$. Does $2A_i$ contain a coset of codimension $O_K(1)$ for some $i$?", "background": "When partitioning a vector space into $K$ parts, at least one part must have substantial additive structure. This problem asks if one piece must have a doubling containing a large coset. It's a partitioning variant of the previous coset problems.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 86, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 116, "problem_number": "GREEN-028", "title": "Gowers Box Norms over Finite Fields", "statement": "Let $p$ be an odd prime and suppose $f : \\mathbb{F}_p^n \\times \\mathbb{F}_p^n \\to \\mathbb{C}$ is bounded pointwise by 1. Suppose $\\mathbb{E}_h \\|\\Delta_{(h,h)}f\\|_\\square^4 \\geq \\delta$. Does $f$ correlate with a function of the form $a(x)b(y)c(x+y)(-1)^{q(x,y)}$?", "background": "This asks for an inverse theorem for a particular Gowers norm in product spaces over finite fields. Understanding which structured functions correlate with high Gowers norm is central to higher-order Fourier analysis.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 84, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 117, "problem_number": "GREEN-029", "title": "Inverse Theorem for Gowers Norms", "statement": "Determine bounds for the inverse theorem for Gowers norms.", "background": "The inverse theorem characterizes functions with large Gowers $U^{s+1}$ norm. Leng-Sah-Sawhney (2024) established a quasi-polynomial inverse theorem for $\\|\\cdot\\|_{U^{s+1}[N]}$ norms for all $s \\geq 3$. Improving to polynomial bounds remains a major challenge in additive combinatorics.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 95, "favorite_count": 6, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 118, "problem_number": "GREEN-030", "title": "Φ(G) and Φ'(G) Coincidence", "statement": "Do $\\Phi(G)$ and $\\Phi'(G)$ coincide?", "background": "This asks whether two different notions of the Frattini-like subgroup of $G$ are equal. The Frattini subgroup consists of non-generators; different definitions can arise in different contexts. Determining their equivalence has implications for group theory.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 4, "view_count": 73, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 119, "problem_number": "GREEN-031", "title": "Sumsets Containing Composites", "statement": "Suppose $A, B \\subset \\{1, \\dots, N\\}$ both have size $N^{0.49}$. Does $A + B$ contain a composite number?", "background": "This asks whether sumsets of moderately large sets must contain composite numbers. Since primes have density $1/\\log N$, sets of size $N^{0.49}$ are much denser, suggesting their sumset should hit composites. However, proving this rigorously requires understanding the additive structure of primes.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 1, "view_count": 81, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 120, "problem_number": "GREEN-032", "title": "Sums of Smooth Numbers", "statement": "Is every $n \\leq N$ the sum of two integers, all of whose prime factors are at most $N^\\varepsilon$?", "background": "Smooth numbers have only small prime factors. This asks if every number is a sum of two smooth numbers, which would show smooth numbers have excellent additive properties. Such a result would have implications for number theory and the distribution of smooth numbers.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 1, "view_count": 88, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 121, "problem_number": "GREEN-033", "title": "Sumsets of Perfect Squares", "statement": "Is there an absolute constant $c > 0$ such that if $A \\subset \\mathbb{N}$ is a set of squares of size at least 2, then $|A + A| \\geq |A|^{1+c}$?", "background": "This asks whether sets of perfect squares have superlinear sumset growth. Squares are highly structured (sparse in $\\mathbb{N}$), and one expects their sumsets to grow substantially. Determining the optimal exponent $c$ is a fundamental problem in additive number theory.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 1, "view_count": 92, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 122, "problem_number": "GREEN-034", "title": "Covering Squares with Sumsets", "statement": "Suppose $A + A$ contains the first $n$ squares. Is $|A| \\geq n^{1-o(1)}$?", "background": "If a set's sumset contains all squares up to $n^2$, must the set have size nearly $n$? This explores the inverse problem: given that a sumset covers a structured set (squares), what can we say about the original set?", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 1, "view_count": 85, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 123, "problem_number": "GREEN-035", "title": "Products of Primes Modulo p", "statement": "Let $p$ be a large prime, and let $A$ be the set of all primes less than $p$. Is every $x \\in \\{1, \\dots, p-1\\}$ congruent to some product $a_1a_2$ modulo $p$?", "background": "This asks whether pairwise products of primes cover all residues modulo $p$. Matom\\\"aki-Ter\\\"av\\\"ainen (2023) made significant progress, showing that products of three primes suffice. Whether two primes suffice remains open.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 1, "view_count": 96, "favorite_count": 6, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 124, "problem_number": "GREEN-036", "title": "Multiplicatively Closed Set Density", "statement": "Let $A$ be the smallest set containing 2 and 3, and closed under the operation $a_1a_2 - 1$ (if $a_1, a_2 \\in A$, then $a_1a_2 - 1 \\in A$). Does $A$ have positive density?", "background": "This defines a set generated by a multiplicative-like operation. Understanding its density is nontrivial because the operation $a_1a_2 - 1$ mixes multiplication with additive structure. Whether such sets have positive density connects number theory with dynamical systems.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 1, "view_count": 77, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 125, "problem_number": "GREEN-037", "title": "Primes with p-2 Having Odd Omega", "statement": "Do there exist infinitely many primes $p$ for which $p-2$ has an odd number of prime factors (counting multiplicity)?", "background": "This asks about the parity of $\\Omega(p-2)$ where $\\Omega(n)$ counts prime factors with multiplicity. Since $p-2$ is even for odd primes $p > 2$, we're asking about the structure of $(p-2)/2$. This is a prime-shifted multiplicative function question.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 1, "view_count": 83, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 126, "problem_number": "GREEN-038", "title": "Difference Sets Containing Squares", "statement": "Is there $c > 0$ such that whenever $A \\subset [N]$ has size $N^{1-c}$, the difference set $A - A$ contains a nonzero square?", "background": "This asks how large a set must be to guarantee its difference set contains a square. Similarly one can ask if $A - A$ contains a prime minus one. These questions probe the additive structure forced by density.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 1, "view_count": 89, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 127, "problem_number": "GREEN-039", "title": "Gaps Between Sums of Two Squares", "statement": "Is there always a sum of two squares between $X - \\frac{1}{10}X^{1/4}$ and $X$?", "background": "Sums of two squares have density $c/\\sqrt{\\log X}$, so gaps can be large. This asks for an upper bound on the largest gap. Such results would improve our understanding of the distribution of representable numbers in quadratic forms.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 1, "view_count": 91, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 128, "problem_number": "GREEN-040", "title": "Waring's Problem Over Finite Fields", "statement": "Determine bounds for Waring's problem over finite fields.", "background": "Waring's problem asks: can every element be written as a sum of $k$ $d$-th powers? Over finite fields $\\mathbb{F}_q$, the problem has different character. Determining the minimum $k$ for given $d$ and $q$ is a classical problem in algebraic number theory.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 1, "view_count": 86, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 129, "problem_number": "GREEN-041", "title": "Cubic Curves in F_p^2", "statement": "Suppose $A \\subset \\mathbb{F}_p^2$ is a set meeting every line in at most 2 points. Is it true that all except $o(p)$ points of $A$ lie on a cubic curve?", "background": "Sets avoiding three collinear points have special structure. Over finite fields, the Hasse-Weil bound and algebraic geometry suggest such sets should lie nearly on a cubic. This is a finite-field analogue of classical incidence geometry.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 6, "view_count": 84, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 130, "problem_number": "GREEN-042", "title": "Collinear Triples and Cubic Curves", "statement": "Fix $k$. Let $A \\subset \\mathbb{R}^2$ be a set of $n$ points with no more than $k$ on any line. Suppose at least $\\delta n^2$ pairs $(x, y) \\in A \\times A$ have the line $xy$ containing a third point of $A$. Is there a cubic curve containing at least $cn$ points of $A$?", "background": "If many pairs determine lines through a third point, the set should have algebraic structure. This asks if a cubic curve captures this structure. It generalizes results from the joints problem and incidence geometry.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 6, "view_count": 78, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 131, "problem_number": "GREEN-043", "title": "Erdős-Szekeres with Visibility", "statement": "Fix integers $k, \\ell$. Given $n \\geq n_0(k, \\ell)$ points in $\\mathbb{R}^2$, is there either a line containing $k$ of them, or $\\ell$ of them that are mutually visible?", "background": "This is a Ramsey-type problem mixing collinearity and visibility (no point blocks the segment between two others). It generalizes the Erdős-Szekeres theorem to a geometric context, asking for unavoidable configurations.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 6, "view_count": 81, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 132, "problem_number": "GREEN-044", "title": "Collinear 4-tuples Force Collinear 5-tuples", "statement": "Suppose $A \\subset \\mathbb{R}^2$ is a set of size $n$ with $cn^2$ collinear 4-tuples. Does it contain 5 points on a line?", "background": "Many collinear 4-tuples suggest the set has strong linear structure. This asks if this forces an actual line through 5 points. It's related to the Szemerédi-Trotter theorem and incidence bounds.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 6, "view_count": 75, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 133, "problem_number": "GREEN-045", "title": "No Three in Line in [N]^2", "statement": "What is the largest subset of the grid $[N]^2$ with no three points on a line? In particular, for $N$ sufficiently large, is it impossible to have a set of size $2N$ with this property?", "background": "The cap set problem in two dimensions. Erdős conjectured sets of size $O(N)$ exist, but proving or disproving this remains open. The problem connects discrete geometry, additive combinatorics, and the polynomial method.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 6, "view_count": 94, "favorite_count": 6, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 134, "problem_number": "GREEN-046", "title": "Smooth Surfaces Intersecting 2-planes", "statement": "Let $\\Gamma$ be a smooth codimension 2 surface in $\\mathbb{R}^n$. Must $\\Gamma$ intersect some 2-dimensional plane in 5 points, if $n$ is sufficiently large?", "background": "This asks about unavoidable intersection patterns between smooth surfaces and planes in high dimensions. It's related to the Kakeya problem and incidence geometry in higher dimensions.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 6, "view_count": 71, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 135, "problem_number": "GREEN-047", "title": "No 5 Points on 2-plane in [N]^d", "statement": "What is the largest subset of $[N]^d$ with no 5 points on a 2-plane?", "background": "This generalizes the no-three-in-line problem to higher dimensions and 2-planes. Determining the maximum size of such sets involves combinatorial geometry and higher-dimensional incidence bounds.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 6, "view_count": 76, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 136, "problem_number": "GREEN-048", "title": "Balanced Ham Sandwich Line", "statement": "Let $X \\subset \\mathbb{R}^2$ be a set of $n$ points. Does there exist a line $\\ell$ through at least two points of $X$ such that the numbers of points on either side of $\\ell$ differ by at most 100?", "background": "This is a variant of the ham sandwich theorem asking for a balanced bisector that passes through points of the set. The classical ham sandwich theorem doesn't require passing through points, making this version more constrained.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 6, "view_count": 79, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 137, "problem_number": "GREEN-049", "title": "Sparse Hitting Set for Rectangles", "statement": "Let $A$ be a set of $n$ points in the plane. Can one select $A' \\subset A$ of size $n/2$ such that any axis-parallel rectangle containing 1000 points of $A$ contains at least one point of $A'$?", "background": "This asks for an efficient hitting set for rectangles defined by a point set. Such results have applications in computational geometry and range searching.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 6, "view_count": 74, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 138, "problem_number": "GREEN-050", "title": "Small Triangles in the Unit Disc", "statement": "Given $n$ points in the unit disc, must there be a triangle of area at most $n^{-2+o(1)}$ determined by them?", "background": "This asks about unavoidable small-area triangles in dense point sets. Cohen-Pohoata-Zakharov (2023) improved the bound to $n^{-8/7-c}$. Reaching the conjectured $n^{-2}$ bound remains open and connects to the Heilbronn triangle problem.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 6, "view_count": 88, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 139, "problem_number": "GREEN-051", "title": "Axis-Parallel Rectangles in Dense Sets", "statement": "Suppose $A$ is an open subset of $[0, 1]^2$ with measure $\\alpha$. Are there four points in $A$ determining an axis-parallel rectangle with area $\\geq c\\alpha^2$?", "background": "This asks if dense sets in the unit square must contain large axis-parallel rectangles. It's a continuous analogue of combinatorial rectangle problems and relates to measure-theoretic ergodic theory.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 6, "view_count": 72, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 140, "problem_number": "GREEN-052", "title": "Equidistribution of Integer Multiples", "statement": "Let $c > 0$ and let $A$ be a set of $n$ distinct integers. Does there exist $\\theta$ such that no interval of length $\\frac{1}{n}$ in $\\mathbb{R}/\\mathbb{Z}$ contains more than $n^c$ of the numbers $\\theta a \\pmod 1$, for $a \\in A$?", "background": "This asks about finding angles $\\theta$ that spread out the set $A$ modulo 1. It's related to discrepancy theory and the distribution of sequences modulo 1.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 1, "view_count": 68, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 141, "problem_number": "GREEN-053", "title": "Random Permutations Fixing k-Sets", "statement": "Let $p(k)$ be the limit as $n \\to \\infty$ of the probability that a random permutation on $[n]$ preserves some set of size $k$. Is $p(k)$ a decreasing function of $k$? Is $p(k) = (C + o(1))k^{-\\alpha}(\\log k)^{-3/2}$ for some absolute constant $C$?", "background": "This concerns the probability that a random permutation fixes some subset. Eberhard-Ford-Green established asymptotic formulas. The question asks for monotonicity and precise asymptotics, connecting combinatorics and probability.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 75, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 142, "problem_number": "GREEN-054", "title": "Comparable Elements in Integer Lattices", "statement": "Consider a set $S \\subset [N]^3$ with the property that any two distinct elements $s, s'$ of $S$ are comparable (in the coordinatewise partial order). Is $|S| \\leq N^{2-\\delta}$ for some $\\delta > 0$?", "background": "An antichain in $[N]^d$ can have size $\\binom{N}{d/2}^d \\sim N^{d/2}$. This asks if totally comparable sets (chains) in 3D are even smaller, achieving $N^{2-\\delta}$ rather than $N^2$. It's a question in extremal combinatorics and poset theory.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 71, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 143, "problem_number": "GREEN-055", "title": "Stable Density on Subspaces", "statement": "Let $A \\subset \\mathbb{F}_2^n$. If $V$ is a subspace, write $\\alpha(V)$ for the density of $A$ on $V$. Is there some $V$ of moderately small codimension on which $\\alpha$ is stable?", "background": "This asks if every set has a subspace where its density doesn't fluctuate wildly. Stability of density on subspaces is fundamental in additive combinatorics over $\\mathbb{F}_2^n$ and relates to the polynomial Freiman-Ruzsa conjecture.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 77, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 144, "problem_number": "GREEN-056", "title": "Almost Invariant Sets Under Affine Maps", "statement": "Suppose $A \\subset \\mathbb{Z}/p\\mathbb{Z}$ has density $\\frac{1}{2}$. Under what conditions on $K$ can $A$ be almost invariant under all maps $\\phi(x) = ax + b$ with $|a|, |b| \\leq K$?", "background": "This asks when a set is nearly preserved under small affine transformations. Understanding which sets have this property connects to additive combinatorics, group actions, and the structure of dense sets in cyclic groups.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 69, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 145, "problem_number": "GREEN-057", "title": "Trace Reconstruction", "statement": "Given a string $x \\in \\{0, 1\\}^n$, let $\\tilde{x}$ be obtained by deleting bits independently at random with probability $\\frac{1}{2}$. How many independent traces $\\tilde{x}_1, \\dots, \\tilde{x}_m$ are needed to reconstruct $x$ with probability 0.9?", "background": "This fundamental problem in computational complexity asks how many noisy observations suffice to recover the original string. Chase (2020) improved bounds to $e^{n^{1/5}\\log^C n}$. Determining the optimal bound is a major open question.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 82, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 146, "problem_number": "GREEN-058", "title": "Irreducibility of Random {0,1} Polynomials", "statement": "Is a random polynomial with coefficients in $\\{0, 1\\}$ and nonzero constant term almost surely irreducible?", "background": "This asks whether most polynomials with binary coefficients are irreducible over $\\mathbb{Q}$. Bary-Soroker-Koukoulopoulos-Kozma (2023) made further progress. The problem connects number theory, probability, and algebraic geometry.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 1, "view_count": 76, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 149, "problem_number": "GREEN-061", "title": "N Queens Problem Asymptotics", "statement": "In how many ways (asymptotically) $Q(n)$ may $n$ non-attacking queens be placed on an $n \\times n$ chessboard?", "background": "The n-queens problem asks for the number of ways to place $n$ queens on an $n \\times n$ board so none attack each other. Determining the asymptotic growth rate of $Q(n)$ is a famous open problem in combinatorics. Rough bounds are known but the exact constant remains elusive.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 94, "favorite_count": 6, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 150, "problem_number": "GREEN-062", "title": "Bounds for Birch's Theorem", "statement": "Let $d \\geq 3$ be odd. Give bounds on $\\nu(d)$ such that if $n > \\nu(d)$ then any homogeneous polynomial $F(\\mathbf{x}) \\in \\mathbb{Z}[x_1, \\dots, x_n]$ of degree $d$ has a nontrivial integer zero.", "background": "Birch's theorem guarantees that homogeneous polynomials of odd degree have nontrivial zeros if there are enough variables. Determining the optimal $\\nu(d)$ is a central problem in Diophantine equations and algebraic number theory.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 1, "view_count": 73, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 151, "problem_number": "GREEN-063", "title": "Solutions to Polynomial Equations in Dense Sets", "statement": "Finding a single solution to $F(x_1, \\dots, x_n) = C$ can be very difficult. What conditions on $A$ ensure that the number of solutions in $A$ is roughly $\\alpha^n$ times the number in $[X]$?", "background": "This asks when a dense set $A$ of density $\\alpha$ contains the \"expected\" number of solutions to a Diophantine equation. Understanding when sparse sets behave like random sets for counting solutions is fundamental in analytic number theory.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 1, "view_count": 70, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 152, "problem_number": "GREEN-064", "title": "Residually Finite Groups", "statement": "Is every group well-approximated by finite groups?", "background": "A group is residually finite if every nontrivial element has a nontrivial image in some finite quotient. This asks if all groups have this property. The answer is known to be no (infinite simple groups), but the question may refer to finitely generated/presented groups, where it remains interesting.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 4, "view_count": 67, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 153, "problem_number": "GREEN-065", "title": "Rado's Boundedness Conjecture", "statement": "Suppose $a_1, \\dots, a_k$ are integers which do not satisfy Rado's condition. Is $c(a_1, \\dots, a_k)$ bounded in terms of $k$ only?", "background": "Rado's condition characterizes which linear equations are partition regular. For equations not satisfying this condition, $c(\\cdot)$ is the minimum number of colors needed to avoid monochromatic solutions. Whether this depends only on $k$ (not the coefficients) is a fundamental question in Ramsey theory.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 72, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 154, "problem_number": "GREEN-066", "title": "Monochromatic x+y and xy", "statement": "If $\\{1, \\dots, N\\}$ is $r$-coloured, then for $N \\geq N_0(r)$ there exist integers $x, y \\geq 3$ such that $x+y$ and $xy$ have the same colour. Find reasonable bounds for $N_0(r)$.", "background": "This asks about unavoidable monochromatic additive-multiplicative patterns. Finding quantitative bounds for $N_0(r)$ connects Ramsey theory with both additive and multiplicative structure.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 78, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 155, "problem_number": "GREEN-067", "title": "Affine Translates of {0,1,3}", "statement": "If $A$ is a set of $n$ integers, what is the maximum number of affine translates of the set $\\{0, 1, 3\\}$ that $A$ can contain?", "background": "This asks how many copies of the pattern $\\{0, 1, 3\\}$ (under affine transformations $x \\mapsto ax + b$) can appear in an $n$-element set. Understanding maximal copies of specific patterns is fundamental in additive combinatorics.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 74, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 156, "problem_number": "GREEN-068", "title": "Restricted Sumsets in Partitions", "statement": "For which values of $k$ is the following true: whenever we partition $[N] = A_1 \\cup \\dots \\cup A_k$, we have $|\\bigcup_{i=1}^k (A_i \\hat{+} A_i)| \\geq \\frac{1}{10} N$?", "background": "This asks how many parts are needed before restricted sumsets (sums of distinct elements) must cover a substantial fraction of $[N]$. The problem connects partition regularity with sumset structure.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 68, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 157, "problem_number": "GREEN-069", "title": "Sum of Cubes in F_3^n", "statement": "Let $A_1, \\dots, A_{100}$ be \"cubes\" in $\\mathbb{F}_3^n$ (images of $\\{0, 1\\}^n$ under linear automorphisms). Is $A_1 + \\dots + A_{100} = \\mathbb{F}_3^n$?", "background": "This asks whether 100 cubes in $\\mathbb{F}_3^n$ always sum to the entire space. It's a question about additive bases and the covering properties of structured sets in vector spaces over finite fields.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 71, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 158, "problem_number": "GREEN-070", "title": "Sets with No Unique Sum Representations", "statement": "What is the size of the smallest set $A \\subset \\mathbb{Z}/p\\mathbb{Z}$ (with at least two elements) for which no element in the sumset $A + A$ has a unique representation?", "background": "This asks for the minimum size of a set where every sum $a + a'$ has multiple representations. Bedert (2023) showed the answer lies between $\\omega(p)\\log p$ and $O(\\log^2 p)$. Closing this gap would deepen our understanding of additive bases.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 76, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 159, "problem_number": "GREEN-071", "title": "Uniform Random Variables with Uniform Sum", "statement": "Suppose $X, Y$ are finitely-supported independent random variables taking integer values such that $X + Y$ is uniformly distributed on its range. Are $X$ and $Y$ themselves uniformly distributed on their ranges?", "background": "This asks if uniform sums force uniform summands. It's a discrete probability question with connections to additive combinatorics and the structure of convolutions.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 70, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 160, "problem_number": "GREEN-072", "title": "Large Subsets of Approximate Groups", "statement": "Suppose $A$ is a $K$-approximate group (not necessarily abelian). Is there $S \\subset A$ with $|S| \\gg K^{-O(1)}|A|$ and $S^8 \\subset A^4$?", "background": "Approximate groups are sets with controlled doubling. This asks if they contain large subsets with even better multiplicative structure. Understanding approximate groups is central to geometric group theory and additive combinatorics.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 4, "view_count": 69, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 161, "problem_number": "GREEN-073", "title": "Structured Subsets with Bounded Doubling", "statement": "Given a set $A \\subset \\mathbb{Z}$ with $D(A) \\leq K$, find a large structured subset $A'$ which \"obviously\" has $D(A') \\leq K + \\varepsilon$.", "background": "Sets with small doubling constant have additive structure. This asks for an explicit, easily verifiable structured subset. Making structure \"obvious\" connects to algorithmic aspects of the Polynomial Freiman-Ruzsa conjecture.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 68, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 162, "problem_number": "GREEN-074", "title": "Sidon Set Size Bounds", "statement": "Write $F(N)$ for the largest Sidon subset of $[N]$. Improve, at least for infinitely many $N$, the bounds $N^{1/2} + O(1) \\leq F(N) \\leq N^{1/2} + N^{1/4} + O(1)$.", "background": "Sidon sets have all pairwise sums distinct. The bounds have been tight for decades. Balogh-Füredi-Roy (2021) obtained a small improvement to the upper bound. Any further progress would be a major breakthrough in additive combinatorics.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 89, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 163, "problem_number": "GREEN-075", "title": "Large Gaps in Dilates", "statement": "Let $p$ be a prime and let $A \\subset \\mathbb{Z}/p\\mathbb{Z}$ be a set of size $\\sqrt{p}$. Is there a dilate of $A$ with a gap of length $100\\sqrt{p}$?", "background": "This asks whether dilates (multiplicative translates) of sets necessarily have large gaps. Understanding the distribution of dilates connects additive and multiplicative combinatorics in cyclic groups.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 72, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 164, "problem_number": "GREEN-076", "title": "Optimal Sidon Bases", "statement": "Are there infinitely many $q$ for which there is a set $A \\subset \\mathbb{Z}/q\\mathbb{Z}$ with $|A| = (\\sqrt{2} + o(1))q^{1/2}$ and $A + A = \\mathbb{Z}/q\\mathbb{Z}$?", "background": "This asks if Sidon-like sets (near-optimal density with few sum collisions) can form additive bases. The coefficient $\\sqrt{2}$ is conjecturally optimal for such constructions.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 75, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 165, "problem_number": "GREEN-077", "title": "Structure of Sets with Bounded Representation", "statement": "Suppose $A \\subset [N]$ has size $\\geq c\\sqrt{N}$ and representation function $r_A(n) \\leq r$ for all $n$. What can be said about the structure of $A$?", "background": "Sets with bounded representation function (few ways to write sums) have special structure. Understanding this structure connects Sidon set theory with additive bases.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 70, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 167, "problem_number": "GREEN-079", "title": "Disjoint Sumsets Construction", "statement": "For arbitrarily large $n$, does there exist an abelian group $H$ with $|H| = n^{2+o(1)}$ and subsets $A_1, \\dots, A_n, B_1, \\dots, B_n$ satisfying $|A_i||B_i| \\geq n^{2-o(1)}$, $|A_i + B_i| = |A_i||B_i|$, such that $A_i + B_i$ are pairwise disjoint from $A_j + B_k$ ($j \\neq k$)?", "background": "This asks if one can partition a group into many disjoint sumsets with no doubling. It connects to the structure of Sidon sets and extremal problems in additive combinatorics.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 69, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 168, "problem_number": "GREEN-080", "title": "Cap Sets in F_7^n", "statement": "What is the largest subset $A \\subset \\mathbb{F}_7^n$ for which $A - A$ intersects $\\{-1, 0, 1\\}^n$ only at 0?", "background": "This is a cap set problem in $\\mathbb{F}_7^n$ with restricted difference set. Recent polynomial method breakthroughs dramatically improved bounds for $\\mathbb{F}_3^n$, but $\\mathbb{F}_7$ remains open.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 73, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 169, "problem_number": "GREEN-081", "title": "Covering by Random Translates", "statement": "If $A \\subset \\mathbb{Z}/p\\mathbb{Z}$ is random with $|A| = \\sqrt{p}$, can we almost surely cover $\\mathbb{Z}/p\\mathbb{Z}$ with $100\\sqrt{p}$ translates of $A$?", "background": "This asks about the covering properties of random sets. Understanding when random sets form good coverings connects probability, additive combinatorics, and coding theory.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 68, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 170, "problem_number": "GREEN-082", "title": "Hamming Ball Covering Growth", "statement": "Let $r$ be fixed and let $H(r)$ be the Hamming ball of radius $r$ in $\\mathbb{F}_2^n$. Let $f(r)$ be the smallest constant such that there exist infinitely many $n$ with subspaces $V_n \\leq \\mathbb{F}_2^n$ satisfying $V_n + H(r) = \\mathbb{F}_2^n$ and $|V_n| = (f(r) + o(1)) \\frac{2^n}{|H(r)|}$. Does $f(r) \\to \\infty$?", "background": "This asks if covering $\\mathbb{F}_2^n$ by Hamming ball translates requires increasingly inefficient packings as $r$ grows. It connects coding theory with the geometry of finite vector spaces.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 2, "view_count": 66, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 171, "problem_number": "GREEN-083", "title": "Pyjama Set Covering", "statement": "How many rotated (about the origin) copies of the \"pyjama set\" $\\{(x, y) \\in \\mathbb{R}^2 : \\operatorname{dist}(x, \\mathbb{Z}) \\leq \\varepsilon\\}$ are needed to cover $\\mathbb{R}^2$?", "background": "The pyjama set is a union of vertical strips. This beautiful geometric problem, solved by Manners (2015), asks how many rotations are needed to cover the plane. It connects geometry, combinatorics, and Fourier analysis.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 6, "view_count": 74, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 172, "problem_number": "GREEN-084", "title": "Cohn-Elkies Scheme for Circle Packings", "statement": "Can the Cohn-Elkies scheme be used to prove the optimal bound for circle-packings?", "background": "Cohn-Elkies developed a linear programming approach that proved optimal sphere packing in dimensions 8 and 24. Whether their method extends to circles in the plane remains a major open question in discrete geometry.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 6, "view_count": 71, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 173, "problem_number": "GREEN-085", "title": "Covering by Residue Classes", "statement": "Let $N$ be large. For each prime $p$ with $N^{0.51} \\leq p < 2N^{0.51}$, pick a residue $a(p) \\in \\mathbb{Z}/p\\mathbb{Z}$. Is $\\#\\{n \\in [N] : n \\equiv a(p) \\pmod p \\text{ for some } p\\} \\gg N^{1-o(1)}$?", "background": "This asks if residue classes from medium-sized primes nearly cover $[N]$. It connects sieve theory with covering problems and the distribution of primes.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 1, "view_count": 69, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 174, "problem_number": "GREEN-086", "title": "Sieving by Many Small Primes", "statement": "Sieve $[N]$ by removing half the residue classes mod $p_i$, for primes $2 \\leq p_1 < p_2 < \\dots < p_{1000} < N^{9/10}$. Does the remaining set have size at most $\\frac{1}{10}N$?", "background": "This asks whether aggressive sieving by many small primes can remove most of $[N]$. Understanding sieve limits is fundamental in analytic number theory.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 1, "view_count": 67, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 175, "problem_number": "GREEN-087", "title": "Residue Class Multiple Coverage", "statement": "Can we pick residue classes $a_p \\pmod p$, one for each prime $p \\leq N$, such that every integer $\\leq N$ lies in at least 10 of them?", "background": "This asks if we can achieve high-multiplicity covering using one residue class per prime. It's dual to sieving problems and connects to the large sieve.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 1, "view_count": 68, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 176, "problem_number": "GREEN-088", "title": "Maximal Covering Interval", "statement": "What is the largest $y$ for which one may cover the interval $[y]$ by residue classes $a_p \\pmod p$, one for each prime $p \\leq x$?", "background": "This is the classical covering problem in sieve theory. Determining the optimal relationship between $x$ and $y$ would have significant implications for understanding the distribution of primes.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 1, "view_count": 70, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 177, "problem_number": "GREEN-089", "title": "Random Walk Mixing on Alternating Groups", "statement": "Pick $x_1, \\dots, x_k \\in A_n$ at random. Is it true that, almost surely as $n \\to \\infty$, the random walk on this set of generators and their inverses equidistributes in time $O(n \\log n)$?", "background": "This asks about mixing time for random walks on the alternating group with random generators. Determining optimal mixing times connects probability, group theory, and spectral graph theory.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 4, "view_count": 69, "favorite_count": 3, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 178, "problem_number": "GREEN-090", "title": "Bounds for Approximate Group Classification", "statement": "Find bounds in the classification theorem for approximate groups.", "background": "The Breuillard-Green-Tao classification shows approximate groups resemble actual groups. However, the bounds in this theorem are extremely poor. Improving them would have significant applications in additive combinatorics and geometric group theory.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 4, "view_count": 72, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 1097, "problem_number": "GREEN-097", "title": "N-Queens Problem Asymptotics", "statement": "In how many ways (asymptotically) $Q(n)$ may $n$ non-attacking queens be placed on an $n \\times n$ chessboard?", "background": "The n-queens problem asks for asymptotic formulas for the number of ways to place n non-attacking queens on an n×n board. Recent work has made progress on both upper and lower bounds, but the precise asymptotic behavior remains elusive.", "difficulty_level_id": 1, "status": "open", "set_id": 3, "category_id": 2, "view_count": 145, "favorite_count": 9, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 1098, "problem_number": "GREEN-098", "title": "Bounds for Homogeneous Polynomial Zeros", "statement": "Let $d \\geq 3$ be an odd integer. Give bounds on $\\nu(d)$ such that if $n > \\nu(d)$ the following is true: given any homogeneous polynomial $F(\\mathbf{x}) \\in \\mathbb{Z}[x_1, \\dots, x_n]$ of degree $d$, there is some $\\mathbf{x} \\in \\mathbb{Z}^n \\setminus \\{\\mathbf{0}\\}$ such that $F(\\mathbf{x}) = 0$.", "background": "This asks for explicit bounds on how many variables are needed to guarantee integer zeros of homogeneous polynomials. Classical results give existence but quantitative bounds remain challenging, especially for higher degrees.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 1, "view_count": 78, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 1099, "problem_number": "GREEN-099", "title": "Polynomial Solutions in Dense Sets", "statement": "Finding a single solution to a polynomial equation $F(x_1, \\dots, x_n) = C$ can be very difficult. What conditions on $A$ ensure that the number of such solutions in $A$ is roughly $\\alpha^n$ times the number of solutions in $[X]$?", "background": "This problem asks when dense sets contain the \"expected\" number of polynomial solutions. Understanding density conditions that guarantee proportional solution counts connects number theory with additive combinatorics.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 1, "view_count": 71, "favorite_count": 4, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 1100, "problem_number": "GREEN-100", "title": "Sofic Groups", "statement": "Is every group well-approximated by finite groups?", "background": "A group is sofic if it can be approximated by finite symmetric groups in a precise sense. Whether all groups are sofic is a major open question in group theory with connections to dynamics, graph theory, and combinatorics.", "difficulty_level_id": 2, "status": "open", "set_id": 3, "category_id": 4, "view_count": 92, "favorite_count": 6, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "set": null, "difficulty": null }, { "id": 1102, "problem_number": "ALG-002", "title": "Hadamard Conjecture", "statement": "For every positive integer $k$, does there exist a Hadamard matrix of order $4k$?", "background": "A Hadamard matrix is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal. The Hadamard conjecture, dating back to 1893, states that such matrices exist for all orders that are multiples of 4. These matrices have important applications in coding theory, signal processing, and quantum information theory. While Hadamard matrices have been constructed for many values of $k$, the smallest order for which existence is unknown is 668. The conjecture has deep connections to combinatorial design theory and remains one of the central problems in discrete mathematics.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 387, "favorite_count": 31, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1103, "problem_number": "ALG-003", "title": "Köthe Conjecture", "statement": "If a ring has no nil ideal other than $\\{0\\}$, does it follow that it has no nil one-sided ideal other than $\\{0\\}$?", "background": "The Köthe conjecture, proposed by Gottfried Köthe in 1930, is a fundamental problem in ring theory concerning the structure of nil ideals. A nil ideal is one in which every element is nilpotent. The conjecture asks whether the absence of two-sided nil ideals implies the absence of one-sided nil ideals. Despite being studied for over 90 years, the problem remains open even for Noetherian rings. The conjecture is related to the Jacobson conjecture and has implications for understanding the structure of general rings. Counterexamples would reveal unexpected asymmetry in the behavior of left and right ideals.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 245, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1104, "problem_number": "ALG-004", "title": "Connes Embedding Problem", "statement": "Can every finite von Neumann algebra be embedded into an ultrapower of the hyperfinite II₁ factor?", "background": "The Connes embedding problem, formulated by Alain Connes in 1976, is a central question in the theory of von Neumann algebras. In 2020, Ji, Natarajan, Vidick, Wright, and Yuen published a paper claiming to have shown the problem has a negative answer, based on connections to quantum complexity theory and the equivalence with Tsirelson's problem in quantum information. However, the problem's status remains subject to verification of their approach. The problem has deep connections to free probability, quantum groups, and mathematical physics, making it one of the most important questions in operator algebra theory.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 312, "favorite_count": 28, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1105, "problem_number": "ALG-005", "title": "Jacobson's Conjecture", "statement": "For a left-and-right Noetherian ring $R$, is the intersection of all powers of the Jacobson radical $J(R)$ equal to zero?", "background": "Jacobson's conjecture addresses a fundamental question about the structure of Noetherian rings. The Jacobson radical of a ring consists of elements that annihilate all simple modules, and understanding its intersection over all powers relates to the ring's nilpotent elements and its representation theory. While the conjecture holds for many important classes of rings (including commutative Noetherian rings), the general case remains open. The problem is closely related to other structural conjectures in non-commutative ring theory, including the Köthe conjecture.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 198, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1106, "problem_number": "ALG-006", "title": "Zauner's Conjecture", "statement": "Do SIC-POVMs (Symmetric Informationally Complete Positive Operator-Valued Measures) exist in all finite dimensions?", "background": "Zauner's conjecture, proposed in 1999, concerns the existence of a special type of quantum measurement in Hilbert spaces of all finite dimensions. A SIC-POVM consists of d² unit vectors in a d-dimensional complex Hilbert space that are equiangular - the absolute inner product of any two distinct vectors is constant. These structures have applications in quantum information theory, quantum state tomography, and quantum cryptography. While SIC-POVMs have been found numerically for all dimensions up to 151 and proven to exist analytically in some special cases, the general existence question remains open. The conjecture has surprising connections to number theory and algebraic geometry.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 176, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1107, "problem_number": "ALG-007", "title": "Casas-Alvero Conjecture", "statement": "If a univariate polynomial $f$ of degree $d$ over a field of characteristic 0 shares a common factor with each of its first $d-1$ derivatives, must $f$ be a power of a linear polynomial?", "background": "The Casas-Alvero conjecture, proposed in 2001, connects the factorization of a polynomial with the factorization of its derivatives. If $f(x)$ has degree $d$ and for each $k = 1, 2, \\ldots, d-1$, the polynomial $f(x)$ shares a root with its $k$-th derivative $f^{(k)}(x)$, the conjecture states that $f$ must be of the form $f(x) = (x - a)^d$ for some constant $a$. While proven for various special cases and low degrees, the general conjecture remains open. It has connections to algebraic geometry and the theory of polynomial equations.", "difficulty_level_id": 3, "status": "open", "category_id": 4, "view_count": 154, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1108, "problem_number": "ALG-008", "title": "Andrews-Curtis Conjecture", "statement": "Can every balanced presentation of the trivial group be transformed into a trivial presentation by a sequence of Nielsen transformations and conjugations of relators?", "background": "The Andrews-Curtis conjecture, proposed in 1965, is a central problem in combinatorial group theory. A balanced presentation has an equal number of generators and relators. The conjecture asks whether such presentations of the trivial group can always be simplified to the form $\\langle x_1, \\ldots, x_n \\mid x_1, \\ldots, x_n \\rangle$ using only Andrews-Curtis moves (Nielsen transformations on relators and conjugations). Despite extensive computational searches and partial results, no counterexample has been found, yet no proof exists. The conjecture has important implications for 3-manifold theory and the classification of homotopy types.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 212, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1109, "problem_number": "ALG-009", "title": "Bounded Burnside Problem", "statement": "For which positive integers $m$ and $n$ is the free Burnside group $B(m,n)$ finite? In particular, is $B(2,5)$ finite?", "background": "The Bounded Burnside problem asks which free Burnside groups are finite. A free Burnside group $B(m,n)$ is the largest group with $m$ generators in which every element has order dividing $n$. It is known that $B(m,n)$ is finite for $n \\in \\{2, 3, 4, 6\\}$ and for certain other special values, and infinite for most large odd exponents. The case $B(2,5)$ has been the subject of extensive computational investigation but remains open. Solving this problem would significantly advance our understanding of periodic groups and torsion in group theory.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1110, "problem_number": "ALG-010", "title": "Herzog-Schönheim Conjecture", "statement": "If a finite system of left cosets of subgroups of a group $G$ partitions $G$, then must at least two of the subgroups have the same index in $G$?", "background": "The Herzog-Schönheim conjecture, proposed in 1974, concerns coset decompositions of groups. If $G$ is partitioned by cosets $g_1H_1, g_2H_2, \\ldots, g_kH_k$ where each $H_i$ is a subgroup of $G$ and the cosets are pairwise disjoint, the conjecture states that at least two of the indices $[G:H_i]$ must be equal. While proven for abelian groups and various other special cases, the general conjecture remains open. It has connections to combinatorial number theory and the structure theory of groups.", "difficulty_level_id": 3, "status": "open", "category_id": 4, "view_count": 142, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1112, "problem_number": "ALG-012", "title": "Existence of Perfect Cuboids", "statement": "Does there exist a rectangular cuboid where all edges, face diagonals, and space diagonals have integer lengths?", "background": "A perfect cuboid (also called a perfect box or Euler brick with space diagonal) would be a rectangular parallelepiped with integer edge lengths $a$, $b$, $c$ such that the face diagonals $\\sqrt{a^2+b^2}$, $\\sqrt{b^2+c^2}$, $\\sqrt{a^2+c^2}$ and the space diagonal $\\sqrt{a^2+b^2+c^2}$ are all integers. Despite extensive computational searches up to very large bounds and numerous partial results, no perfect cuboid has been found, nor has impossibility been proven. The problem has connections to Diophantine equations and elliptic curves, and has fascinated both amateur and professional mathematicians for centuries.", "difficulty_level_id": 3, "status": "open", "category_id": 4, "view_count": 234, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1114, "problem_number": "ALG-014", "title": "McKay Conjecture", "statement": "For a finite group $G$ and prime $p$, is the number of irreducible complex characters of $G$ whose degree is not divisible by $p$ equal to the corresponding number for the normalizer of a Sylow $p$-subgroup?", "background": "The McKay conjecture, proposed in the 1970s, is a central problem in the representation theory of finite groups. It predicts a surprising relationship between the character degrees of a group and those of a much smaller subgroup (the normalizer of a Sylow $p$-subgroup). The conjecture has been verified for many important classes of groups and has led to deep insights about the structure of character tables. A proof was announced in 2007 by Isaacs, Malle, and Navarro assuming the classification of finite simple groups, though subtle gaps in the argument have led to ongoing refinement of the proof.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 156, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1115, "problem_number": "ALG-015", "title": "Are All Groups Surjunctive?", "statement": "Is every group surjunctive? That is, for any group $G$, if $\\phi: A^G \\to A^G$ is a cellular automaton that is injective, must it also be surjective?", "background": "A group $G$ is called surjunctive if every injective cellular automaton on $G$ is automatically surjective. Equivalently, this asks whether the dynamical system defined by a cellular automaton on the group can be injective without being bijective. Gromov and Weiss proved that all sofic groups are surjunctive, and all known groups are sofic, but it remains unknown whether all groups are surjunctive. The question has deep connections to symbolic dynamics, geometric group theory, and the Garden of Eden theorem from cellular automaton theory. A negative answer would be quite surprising and would reveal fundamental limitations in our understanding of group actions.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 143, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1119, "problem_number": "NT-016", "title": "Catalan-Mersenne Conjecture", "statement": "Are all Catalan-Mersenne numbers $C_n$ composite for $n > 4$? Here $C_0 = 2$ and $C_{n+1} = 2^{C_n} - 1$.", "background": "The Catalan-Mersenne conjecture concerns a doubly exponential sequence where each term is a Mersenne number with exponent equal to the previous term. The sequence grows extraordinarily rapidly: $C_0 = 2$, $C_1 = 3$, $C_2 = 7$, $C_3 = 127$, $C_4 = 170141183460469231731687303715884105727$ (a 39-digit number). The first four terms are prime, but $C_5$ has over $10^{38}$ digits, making it far beyond reach of current computational methods. The conjecture predicts that all subsequent terms are composite. This problem connects to deep questions about the distribution of Mersenne primes and the limitations of our ability to determine primality for extremely large numbers.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 287, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1120, "problem_number": "NT-017", "title": "Are There Infinitely Many Mersenne Primes?", "statement": "Are there infinitely many prime numbers of the form $2^p - 1$ where $p$ is prime?", "background": "Mersenne primes are primes of the form $M_p = 2^p - 1$ where $p$ is itself prime. As of 2024, only 51 Mersenne primes are known, the largest being $2^{82589933} - 1$ discovered in 2018. Despite their rarity, it is conjectured that infinitely many exist. Mersenne primes are intimately connected to perfect numbers through the Euclid-Euler theorem: an even number is perfect if and only if it has the form $2^{p-1}(2^p-1)$ where $2^p-1$ is a Mersenne prime. The question of whether infinitely many Mersenne primes exist is closely related to our understanding of the distribution of primes and has implications for both pure and applied mathematics, as Mersenne primes are used in pseudorandom number generation and cryptography.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 567, "favorite_count": 49, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1121, "problem_number": "GEO-001", "title": "Sphere Packing Problem in Higher Dimensions", "statement": "What is the densest packing of spheres in dimensions 4 through 23? More generally, what is the optimal sphere packing density in dimension $n$?", "background": "The sphere packing problem asks for the densest arrangement of non-overlapping spheres in $n$-dimensional space. In dimension 3, Kepler's conjecture (proved by Hales in 1998) shows the densest packing has density $\\pi/\\sqrt{18} \\approx 0.7405$. In 2016, Maryna Viazovska proved that the E₈ lattice gives the densest packing in dimension 8, and shortly after, Cohn, Kumar, Miller, Radchenko, and Viazovska proved the Leech lattice is optimal in dimension 24. However, dimensions 4-7 and 9-23 remain open, as do almost all higher dimensions. The problem has deep connections to coding theory, number theory, and optimization.", "difficulty_level_id": 5, "status": "open", "category_id": 6, "view_count": 398, "favorite_count": 34, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1122, "problem_number": "GEO-002", "title": "Mahler's Conjecture", "statement": "Among all centrally symmetric convex bodies in $\\mathbb{R}^n$, does the cube (or cross-polytope) minimize the product of the body's volume and the volume of its polar dual?", "background": "Mahler's conjecture, proposed in 1939, concerns a fundamental geometric quantity called the Mahler volume, defined as the product of a convex body's volume with the volume of its polar dual. Kurt Mahler conjectured that among all centrally symmetric convex bodies in $\\mathbb{R}^n$, this product is minimized by the cube and the cross-polytope (which are dual to each other). The conjecture has been proved in dimension 2 by Mahler himself, and partial results exist for special classes of bodies, but the general case remains open. The problem connects convex geometry, functional analysis, and the theory of Banach spaces.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 245, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1123, "problem_number": "GEO-003", "title": "The Illumination Conjecture", "statement": "Can every convex body in $n$-dimensional space be illuminated by at most $2^n$ point light sources?", "background": "The illumination conjecture, also known as Hadwiger's covering conjecture in one of its forms, asks whether every convex body in $\\mathbb{R}^n$ can be illuminated by at most $2^n$ point light sources placed outside the body. A point on the surface is considered illuminated if the ray from the light source to that point does not pass through the interior of the body. The conjecture has been proven for $n = 2$ and $n = 3$, but remains open for higher dimensions. The problem is closely related to covering problems and has connections to discrete geometry and combinatorics.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 187, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1124, "problem_number": "GEO-004", "title": "Kakeya Needle Problem", "statement": "What is the minimum area of a region in the plane in which a unit line segment can be continuously rotated through 360 degrees?", "background": "The Kakeya needle problem asks for the smallest area set in the plane within which a unit line segment can be rotated continuously through 360 degrees, returning to its initial position. While Besicovitch showed in 1928 that there exist Kakeya sets of arbitrarily small positive measure, the question of what happens when we require the set to be connected or simply connected remains fascinating. In higher dimensions, the Kakeya conjecture (related but distinct) concerns sets containing unit line segments in every direction and has deep connections to harmonic analysis, partial differential equations, and number theory. The finite field analog was resolved by Dvir in 2008 using the polynomial method.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 312, "favorite_count": 27, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1125, "problem_number": "GEO-005", "title": "Bellman's Lost in a Forest Problem", "statement": "What is the shortest path that guarantees escape from a forest of known shape and size, starting from an unknown location?", "background": "Bellman's lost in a forest problem asks for the shortest universal path that guarantees reaching the boundary of a region, regardless of starting position and orientation. For a circular forest of radius 1, the problem was solved by various authors with a path of length approximately 7.2898. However, for other shapes like squares or equilateral triangles, the optimal escape path remains unknown. This problem has applications to robotics, search and rescue operations, and computational geometry. It connects to questions about curve shortening, geometric optimization, and worst-case analysis in motion planning.", "difficulty_level_id": 3, "status": "open", "category_id": 6, "view_count": 198, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1131, "problem_number": "COMB-003", "title": "The Union-Closed Sets Conjecture", "statement": "For any finite family of finite sets that is closed under taking unions, must there exist an element that belongs to at least half of the sets?", "background": "The union-closed sets conjecture, also known as Frankl's conjecture after Peter Frankl who popularized it in 1979, is a simple-to-state problem in extremal combinatorics. A family of sets is union-closed if the union of any two sets in the family is also in the family. The conjecture asserts that in any non-trivial union-closed family, some element appears in at least half of the sets. Despite extensive research and verification for small cases, the general conjecture remains open. It has connections to lattice theory, Boolean functions, and information theory.", "difficulty_level_id": 4, "status": "open", "category_id": 2, "view_count": 334, "favorite_count": 28, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1132, "problem_number": "COMB-004", "title": "Singmaster's Conjecture", "statement": "Does there exist a finite upper bound on how many times a number (other than 1) can appear in Pascal's triangle?", "background": "Singmaster's conjecture, proposed by David Singmaster in 1971, concerns the frequency of entries in Pascal's triangle. While 1 appears infinitely often (along the edges), and 2 appears exactly three times, larger numbers can appear multiple times in different positions. For example, 120 appears six times. The conjecture states that there exists an absolute constant $C$ such that no number appears more than $C$ times in Pascal's triangle (excluding 1). Singmaster himself proved that the number of occurrences is at most $O(\\log n / \\log \\log n)$ for the entry $n$. The conjecture connects to Diophantine equations and the distribution of binomial coefficients.", "difficulty_level_id": 3, "status": "open", "category_id": 2, "view_count": 298, "favorite_count": 26, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1135, "problem_number": "SET-001", "title": "The Continuum Hypothesis", "statement": "Is there a set whose cardinality is strictly between that of the integers and the real numbers?", "background": "The continuum hypothesis (CH), proposed by Georg Cantor in 1878, states that there is no set with cardinality strictly between that of the integers and the real numbers. Equivalently, it asserts that the cardinality of the continuum (the real numbers) is $\\aleph_1$, the second smallest infinite cardinal. Gödel proved in 1940 that CH is consistent with ZFC (if ZFC is consistent), and Cohen proved in 1963 that the negation of CH is also consistent with ZFC. Thus, CH is independent of the standard axioms of set theory. This means CH can neither be proved nor disproved from ZFC alone, making it one of the most philosophically significant results in mathematical logic.", "difficulty_level_id": 5, "status": "open", "category_id": 13, "view_count": 623, "favorite_count": 54, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1137, "problem_number": "NT-019", "title": "Are There Infinitely Many Sophie Germain Primes?", "statement": "Are there infinitely many primes $p$ such that $2p + 1$ is also prime?", "background": "A Sophie Germain prime is a prime $p$ where $2p+1$ is also prime. These primes are named after French mathematician Sophie Germain, who used them in her work on Fermat's Last Theorem. Examples include 2, 3, 5, 11, 23, and 29. The conjecture that infinitely many exist is closely related to the twin prime conjecture and is similarly difficult. Sophie Germain primes have applications in cryptography and are used in some primality testing algorithms. As of 2024, the largest known Sophie Germain prime has over 400,000 digits. The problem remains one of the major open questions about prime distribution.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 389, "favorite_count": 33, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1138, "problem_number": "AG-001", "title": "The Hodge Conjecture", "statement": "On a projective algebraic variety, is every Hodge class a rational linear combination of classes of algebraic cycles?", "background": "The Hodge conjecture is one of the seven Millennium Prize Problems, with a $1 million prize for its solution. Proposed by William Hodge in 1950, it concerns the deep relationship between the topology and algebraic geometry of complex projective varieties. In simple terms, it asks whether certain topological cycles (Hodge classes) can be represented by algebraic cycles (subvarieties). The conjecture has been verified in many special cases, including for curves, surfaces, and abelian varieties, but the general case remains stubbornly open. It connects algebraic geometry, topology, and complex analysis in profound ways.", "difficulty_level_id": 5, "status": "open", "category_id": 5, "view_count": 534, "favorite_count": 46, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1140, "problem_number": "AG-003", "title": "The Birch and Swinnerton-Dyer Conjecture", "statement": "For an elliptic curve $E$ over the rationals, does the rank of its group of rational points equal the order of vanishing of its $L$-function at $s=1$?", "background": "The Birch and Swinnerton-Dyer (BSD) conjecture is one of the seven Millennium Prize Problems. It connects the arithmetic properties of elliptic curves (specifically, the group of rational points) with analytic properties (the behavior of the associated $L$-function). The conjecture predicts a precise relationship between these seemingly disparate aspects. It has been verified computationally for millions of curves and proven in special cases, but the general conjecture remains open. A proof would revolutionize our understanding of elliptic curves and have applications to cryptography and number theory. The conjecture also relates to the Langlands program and modern arithmetic geometry.", "difficulty_level_id": 5, "status": "open", "category_id": 5, "view_count": 687, "favorite_count": 59, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1141, "problem_number": "DYN-001", "title": "The Weinstein Conjecture", "statement": "Does every Reeb vector field on a closed contact manifold have at least one periodic orbit?", "background": "The Weinstein conjecture, proposed by Alan Weinstein in 1978, is a fundamental problem in symplectic geometry and dynamical systems. A Reeb vector field is a special type of vector field on a contact manifold, and the conjecture predicts the existence of closed orbits under very general conditions. The conjecture was proven in dimension 3 by Hofer in 1993 using pseudoholomorphic curves, and has been established in many other cases. However, the general case remains open. The conjecture has deep connections to Hamiltonian dynamics, celestial mechanics, and the study of periodic phenomena in physics.", "difficulty_level_id": 4, "status": "open", "category_id": 11, "view_count": 276, "favorite_count": 23, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1142, "problem_number": "DYN-002", "title": "The Painlevé Conjecture", "statement": "In the $n$-body problem with $n \\geq 4$, can non-collision singularities occur in finite time?", "background": "The Painlevé conjecture concerns the $n$-body problem in celestial mechanics, asking whether the motion of $n$ point masses under gravitational attraction can develop a singularity (infinite velocities or unbounded positions) in finite time without any collisions occurring. For $n=3$, Sundman proved in 1912 that non-collision singularities cannot occur, but for $n \\geq 4$ the question remains open. Xia constructed examples showing that certain types of unbounded behavior are possible, but the existence of true non-collision singularities (where velocities become infinite) remains unproven. The problem has implications for the long-term behavior of planetary systems and the foundations of classical mechanics.", "difficulty_level_id": 5, "status": "open", "category_id": 11, "view_count": 298, "favorite_count": 25, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1144, "problem_number": "GT-008", "title": "Cereceda's Conjecture", "statement": "For any $k$-chromatic graph, can its $k$-colorings be transformed into each other by recoloring one vertex at a time, staying within $k$ colors, in polynomial time in the number of vertices?", "background": "Cereceda's conjecture concerns the diameter of the reconfiguration graph of $k$-colorings. Given a graph $G$ with chromatic number $k$, consider the graph whose vertices are all proper $k$-colorings of $G$, with two colorings adjacent if they differ on exactly one vertex. The conjecture, proposed in 2007, states that this graph has diameter at most $O(n^2)$ where $n$ is the number of vertices in $G$. The problem is motivated by questions in computational complexity and has connections to mixing times of Markov chains. While progress has been made for special graph classes, the general conjecture remains open.", "difficulty_level_id": 3, "status": "open", "category_id": 3, "view_count": 198, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1146, "problem_number": "TOP-003", "title": "The Whitehead Conjecture", "statement": "Is every aspherical closed manifold whose fundamental group has no non-trivial perfect normal subgroups a $K(\\pi, 1)$ space?", "background": "The Whitehead conjecture, posed by J.H.C. Whitehead, concerns a special class of topological spaces. A space is aspherical if all its homotopy groups above dimension 1 vanish, and it is a $K(\\pi, 1)$ if it is aspherical and connected. The conjecture asks whether certain algebraic conditions on the fundamental group guarantee the topological property of being aspherical. The Poincaré conjecture can be viewed as a special case. While the conjecture is known to hold for many important classes of manifolds, including those with non-positive sectional curvature, the general case remains open. The problem connects algebraic topology, geometric topology, and group theory.", "difficulty_level_id": 4, "status": "open", "category_id": 14, "view_count": 234, "favorite_count": 20, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1149, "problem_number": "GEO-006", "title": "The Knaster Problem", "statement": "Can a solid cube be completely covered by finitely many smaller homothetic cubes with ratio less than 1, such that the interiors are disjoint?", "background": "The Knaster problem, also known as the cube packing problem, asks whether a unit cube can be covered by finitely many non-overlapping smaller cubes, each similar to the original with ratio $< 1$. In 1979, Mycielski proved this is impossible in dimension 2 (for squares), but the 3-dimensional case remains open. The problem has connections to measure theory, geometric covering problems, and Banach-Tarski-like paradoxes. A positive answer would be quite surprising as it would demonstrate a counterintuitive property of 3-dimensional space. The problem has inspired research into covering and packing problems in higher dimensions.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 189, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1153, "problem_number": "NT-022", "title": "Polignac's Conjecture", "statement": "For every even number $n$, are there infinitely many pairs of consecutive primes differing by $n$?", "background": "Polignac's conjecture, proposed by Alphonse de Polignac in 1849, is a vast generalization of the twin prime conjecture. It asserts that for every even integer $n$, there exist infinitely many prime gaps of exactly size $n$. The twin prime conjecture is the special case $n = 2$. While Zhang's 2013 breakthrough showed infinitely many bounded gaps exist, proving the existence of infinitely many gaps of any specific even size remains open. The conjecture relates to the Hardy-Littlewood conjectures and our understanding of the distribution of primes. Even proving the existence of infinitely many prime gaps of size 6 would be a major breakthrough.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 389, "favorite_count": 33, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1156, "problem_number": "ALG-016", "title": "The Babai Conjecture on Graph Isomorphism", "statement": "Can graph isomorphism be decided in quasi-polynomial time for all graphs?", "background": "The Babai conjecture concerns the computational complexity of determining whether two graphs are isomorphic. In 2015, László Babai announced a quasi-polynomial time algorithm for graph isomorphism (running in time $2^{O(\\log^c n)}$ for some constant $c$), improving on the previous best bound. While a flaw was found in the original proof, Babai repaired it in 2017. However, whether graph isomorphism is in P (polynomial time) remains open. The problem sits in NP but is not known to be NP-complete, occupying a special place in complexity theory. The resolution has important implications for cryptography and the structure of complexity classes.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 445, "favorite_count": 38, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1157, "problem_number": "NT-023", "title": "Pillai's Conjecture", "statement": "For each positive integer $k$, does the equation $|2^m - 3^n| = k$ have only finitely many solutions in positive integers $m$ and $n$?", "background": "Pillai's conjecture, proposed by Subbayya Sivasankaranarayana Pillai, concerns the gaps between powers of 2 and powers of 3. The conjecture generalizes to any two multiplicatively independent integers $a$ and $b$: the equation $|a^m - b^n| = k$ should have only finitely many solutions for each fixed $k$. This is related to the abc conjecture and to understanding the distribution of exponential Diophantine equations. The conjecture has been proven for many special cases but remains open in general. It connects to transcendental number theory and the study of linear forms in logarithms.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 198, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1158, "problem_number": "NT-024", "title": "Erdős-Straus Conjecture", "statement": "For every integer $n \\geq 2$, can $\\frac{4}{n}$ be expressed as the sum of three unit fractions $\\frac{1}{x} + \\frac{1}{y} + \\frac{1}{z}$?", "background": "The Erdős-Straus conjecture asks whether every fraction $4/n$ (for $n \\geq 2$) can be written as a sum of three unit fractions (fractions with numerator 1). The conjecture has been verified computationally for all $n$ up to $10^{17}$ and proven for several infinite families of values, but the general case remains open. Egyptian fraction representations have been studied since ancient times, and this problem connects to number theory, combinatorics, and computational mathematics. Related problems concern representing other fractions as sums of unit fractions with various restrictions.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 289, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1159, "problem_number": "NT-025", "title": "The Gauss Circle Problem", "statement": "What is the optimal error term in the formula for the number of lattice points inside a circle of radius $r$?", "background": "The Gauss circle problem asks for the number of integer lattice points $(x,y)$ satisfying $x^2 + y^2 \\leq r^2$. The main term is $\\pi r^2$ (the area of the circle), but determining the optimal error term has been a central problem in analytic number theory for over 150 years. It is known that the error is $O(r^{2/3})$ and conjectured to be $O(r^{1/2 + \\epsilon})$ for any $\\epsilon > 0$, but this has not been proven. The problem connects to the distribution of lattice points, the theory of the Riemann zeta function, and has inspired numerous techniques in analytic number theory.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 367, "favorite_count": 31, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1160, "problem_number": "ALG-017", "title": "Birch-Tate Conjecture", "statement": "Does the order of the center of the Steinberg group of the ring of integers of a number field relate to the value of the Dedekind zeta function at $s=-1$?", "background": "The Birch-Tate conjecture concerns the relationship between algebraic K-theory and special values of zeta functions. Specifically, it relates the order of $K_2$ of the ring of integers of a number field to the value of the Dedekind zeta function at $s = -1$. This conjecture is part of a broader program connecting algebraic K-theory to number theory and has been verified in many special cases. It generalizes ideas from class field theory and has connections to the Lichtenbaum conjectures. The problem sits at the intersection of algebraic number theory, algebraic K-theory, and the theory of zeta functions.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 178, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1161, "problem_number": "ALG-018", "title": "Hilbert's Fifteenth Problem", "statement": "Can Schubert calculus be given a rigorous foundation?", "background": "Hilbert's fifteenth problem, from his famous 1900 list, asks for a rigorous foundation of Schubert's enumerative calculus. Schubert calculus is a method for solving problems in enumerative geometry, such as counting the number of lines in 3-space that meet four given lines. While modern algebraic geometry has provided substantial progress through intersection theory and the development of Chow rings, aspects of the problem remain active areas of research. The development of Gromov-Witten invariants and quantum cohomology has provided new tools, but questions about the complete rigor of classical Schubert calculus in all dimensions continue to be investigated.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 245, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1162, "problem_number": "ALG-019", "title": "Hilbert's Sixteenth Problem", "statement": "What is the maximum number and relative positions of limit cycles for polynomial vector fields of degree $n$ in the plane?", "background": "Hilbert's sixteenth problem consists of two parts. The first part (topology of algebraic curves) asks about the possible configurations of connected components of real algebraic curves. The second part asks for the maximum number and possible configurations of limit cycles of polynomial vector fields of degree $n$ in the plane. This second part remains largely open even for $n=2$ (quadratic systems). The problem is fundamental to the qualitative theory of differential equations and has applications to dynamical systems, control theory, and mathematical biology. Despite over a century of research, even basic questions about quadratic systems remain unresolved.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 312, "favorite_count": 27, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1163, "problem_number": "GEO-008", "title": "The Inscribed Square Problem", "statement": "Does every simple closed curve in the plane contain four points that form the vertices of a square?", "background": "The inscribed square problem, also known as Toeplitz' conjecture, asks whether every Jordan curve (simple closed curve) in the plane contains four points forming a square. The problem has been open since 1911. It is known to be true for smooth curves and for many other special cases, but the general case for arbitrary continuous curves remains unproven. The problem is related to other inscribed polygon problems and has connections to topology, dynamical systems, and geometric measure theory. Even proving the existence of an inscribed rectangle with sides in a given ratio remains challenging for general curves.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 456, "favorite_count": 39, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1164, "problem_number": "GEO-009", "title": "Falconer's Conjecture", "statement": "If a compact set in $\\mathbb{R}^d$ has Hausdorff dimension greater than $d/2$, must it determine a set of distances with positive Lebesgue measure?", "background": "Falconer's conjecture concerns the relationship between the fractal dimension of a set and the set of distances between its points. Proposed by Kenneth Falconer in 1985, it states that if a compact set $E \\subset \\mathbb{R}^d$ has Hausdorff dimension strictly greater than $d/2$, then the distance set $\\{|x-y| : x, y \\in E\\}$ has positive Lebesgue measure. The conjecture has been proven in dimension 2 but remains open in higher dimensions. It has deep connections to harmonic analysis, geometric measure theory, and additive combinatorics. Recent progress using polynomial methods has improved bounds but not resolved the conjecture.", "difficulty_level_id": 5, "status": "open", "category_id": 6, "view_count": 289, "favorite_count": 25, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1165, "problem_number": "GT-010", "title": "The Total Coloring Conjecture", "statement": "Can every graph be totally colored with at most $\\Delta + 2$ colors, where $\\Delta$ is the maximum degree?", "background": "The total coloring conjecture, proposed independently by Behzad and Vizing in the 1960s, concerns coloring both vertices and edges of a graph such that no two adjacent or incident elements receive the same color. The conjecture states that every graph can be totally colored using at most $\\Delta + 2$ colors where $\\Delta$ is the maximum degree. Vizing proved that at most $\\Delta + 2$ colors suffice, and it is trivial that at least $\\Delta + 1$ are needed. The conjecture asks whether the upper bound is tight. It has been verified for many graph classes but remains open in general. The problem has applications to scheduling and resource allocation.", "difficulty_level_id": 3, "status": "open", "category_id": 3, "view_count": 234, "favorite_count": 20, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1171, "problem_number": "NT-026", "title": "The Odd Perfect Number Conjecture", "statement": "Do there exist any odd perfect numbers? (A perfect number equals the sum of its proper divisors.)", "background": "A perfect number is a positive integer that equals the sum of its proper positive divisors. Euclid showed that numbers of the form $2^{p-1}(2^p - 1)$ are perfect when $2^p - 1$ is prime (Mersenne prime), giving all known even perfect numbers. Whether odd perfect numbers exist has been an open question for over 2000 years. It is known that if an odd perfect number exists, it must be greater than $10^{1500}$, have at least 101 prime factors, and satisfy numerous other constraints. The problem connects to prime number theory, divisibility, and has inspired extensive computational searches. Most mathematicians believe no odd perfect numbers exist.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 678, "favorite_count": 58, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1172, "problem_number": "NT-027", "title": "Firoozbakht's Conjecture", "statement": "Is the sequence $p_n^{1/n}$ strictly decreasing, where $p_n$ is the $n$-th prime?", "background": "Firoozbakht's conjecture, proposed in 1982, states that the sequence $(p_n)^{1/n}$ is strictly decreasing, where $p_n$ denotes the $n$-th prime number. This is equivalent to saying that $p_{n+1}^n < p_n^{n+1}$ for all $n$. The conjecture is stronger than Cramér's conjecture about prime gaps and has been verified computationally for all primes up to very large values. If true, it would imply strong results about the distribution of primes and prime gaps. The conjecture remains open despite extensive numerical evidence supporting it.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 198, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1173, "problem_number": "AG-004", "title": "The Tate Conjecture", "statement": "For varieties over finite fields, are the $\\ell$-adic representations arising from étale cohomology related to algebraic cycles in the expected way?", "background": "The Tate conjecture, proposed by John Tate in 1963, is a fundamental problem in arithmetic geometry. It concerns the relationship between algebraic cycles on algebraic varieties over finite fields and the Galois representations arising from étale cohomology. The conjecture would provide a powerful tool for understanding rational equivalence of cycles. It has been proven for divisors (codimension 1 cycles) on abelian varieties and for various other special cases. The conjecture is closely related to the Hodge conjecture and the Birch and Swinnerton-Dyer conjecture, forming part of a web of deep conjectures in arithmetic geometry.", "difficulty_level_id": 5, "status": "open", "category_id": 5, "view_count": 256, "favorite_count": 22, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1176, "problem_number": "SET-002", "title": "Suslin's Problem", "statement": "If a dense linear order without endpoints is complete and has the countable chain condition, must it be isomorphic to the real numbers?", "background": "Suslin's problem, posed by Mikhail Suslin in 1920, asks whether the real numbers can be characterized by certain order-theoretic properties. Specifically, it asks if every complete dense linear order without endpoints satisfying the countable chain condition (every family of disjoint open intervals is countable) must be order-isomorphic to $\\mathbb{R}$. In 1967, it was shown that this question is independent of ZFC set theory - both positive and negative answers are consistent with the standard axioms. A \"Suslin line\" (a counterexample) exists in some models of set theory but not in others. The problem inspired fundamental developments in set theory and the study of independence results.", "difficulty_level_id": 5, "status": "open", "category_id": 13, "view_count": 289, "favorite_count": 25, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1179, "problem_number": "NT-028", "title": "Schinzel's Hypothesis H", "statement": "If polynomials satisfy certain necessary divisibility conditions, do they simultaneously produce infinitely many primes for integer inputs?", "background": "Schinzel's Hypothesis H is a sweeping generalization of many conjectures about primes, including the twin prime conjecture, Sophie Germain prime conjecture, and Dickson's conjecture. It states that if $f_1, \\ldots, f_k$ are irreducible polynomials with integer coefficients and positive leading coefficients, and no prime divides all values $f_1(n) \\cdots f_k(n)$ simultaneously for all integers $n$, then there are infinitely many integers $n$ for which all $f_i(n)$ are prime. If true, it would unify and resolve numerous open problems about prime-producing polynomials. The hypothesis remains wide open despite its fundamental importance.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 298, "favorite_count": 26, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1180, "problem_number": "ALG-020", "title": "The Uniform Boundedness Conjecture", "statement": "Is there a bound $B(g, d)$ such that every curve of genus $g$ over a number field of degree $d$ has at most $B(g, d)$ rational points?", "background": "The uniform boundedness conjecture for rational points on curves asks whether, for fixed genus $g$ and degree $d$, there exists a bound on the number of rational points on genus-$g$ curves over number fields of degree $d$. This would be a vast generalization of Mordell's conjecture (now Faltings' theorem, which shows finiteness but not uniform bounds). The conjecture has been proven for $g = 1$ (elliptic curves) by Mazur and Merel, but remains open for $g \\geq 2$. It connects to deep questions in arithmetic geometry about the distribution of rational points on varieties.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 234, "favorite_count": 20, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1181, "problem_number": "ALG-021", "title": "The Pierce-Birkhoff Conjecture", "statement": "Is every piecewise-polynomial function $f: \\mathbb{R}^n \\to \\mathbb{R}$ the maximum of finitely many minimums of finite collections of polynomials?", "background": "The Pierce-Birkhoff conjecture asks whether every continuous piecewise polynomial function on $\\mathbb{R}^n$ can be represented using only the operations of addition, multiplication, and taking finite suprema and infima of polynomial functions. The conjecture has been verified in dimension 1 and for $n = 2$ in special cases, but remains open for general $n \\geq 2$. The problem has connections to real algebraic geometry, approximation theory, and constructive mathematics. A positive answer would provide powerful representation theorems for piecewise-defined functions.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 178, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1182, "problem_number": "ALG-022", "title": "Serre's Positivity Conjecture", "statement": "If $R$ is a regular local ring and $P, Q$ are prime ideals with intersecting dimensions satisfying a certain condition, is the intersection multiplicity positive?", "background": "Serre's positivity conjecture (part of Serre's multiplicity conjectures) concerns intersection multiplicities in commutative algebra. It states that if $R$ is a commutative regular local ring and $P, Q$ are prime ideals with $\\dim(R/P) + \\dim(R/Q) = \\dim(R)$, then the intersection multiplicity $\\chi(R/P, R/Q) > 0$. The conjecture was proven by Gabber, Paul Roberts, and others in the 1980s for rings containing a field, but remains open in mixed characteristic (characteristic 0 with positive characteristic residue field). The problem connects to algebraic K-theory and has applications to intersection theory.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 156, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1183, "problem_number": "NT-029", "title": "Artin's Conjecture on Primitive Roots", "statement": "For how many prime numbers $p$ is a given integer $a$ (not $\\pm 1$ or a perfect square) a primitive root modulo $p$?", "background": "Artin's conjecture on primitive roots states that any integer $a$ that is neither $-1$, $\\pm 1$, nor a perfect square is a primitive root modulo infinitely many primes, and gives a conjectured density for such primes. For example, it predicts that 2 is a primitive root for approximately 37.4% of all primes. Under the assumption of the generalized Riemann hypothesis, Hooley proved the conjecture in 1967. However, the unconditional case remains open. The conjecture has important implications for the distribution of generators in finite fields and connects to class field theory.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 267, "favorite_count": 23, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1184, "problem_number": "NT-030", "title": "The abc Conjecture", "statement": "For coprime integers $a, b, c$ with $a + b = c$, is $c$ usually not much larger than the product of distinct primes dividing $abc$?", "background": "The abc conjecture, proposed by Oesterlé and Masser in 1985, is one of the most important open problems in number theory. It states that for any $\\epsilon > 0$, there are only finitely many triples of coprime positive integers $(a,b,c)$ with $a + b = c$ such that $c > \\text{rad}(abc)^{1+\\epsilon}$, where $\\text{rad}(n)$ is the product of distinct prime factors of $n$. If true, it would imply Fermat's Last Theorem, Mordell's conjecture (already proven), and many other results. Shinichi Mochizuki claimed a proof in 2012 using inter-universal Teichmüller theory, but the mathematical community has not reached consensus on its validity.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 892, "favorite_count": 76, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1185, "problem_number": "GEO-010", "title": "The Shephard's Problem", "statement": "Can the unit ball in $\\mathbb{R}^n$ be illuminated by fewer than $2^n$ directions?", "background": "Shephard's problem, a variant of the illumination problem, asks how many directions are needed to illuminate the entire boundary of the unit ball in $n$-dimensional space. A direction illuminates a boundary point if moving in that direction from the point leads outside the ball. It is known that $2^n$ directions suffice (by considering all combinations of positive/negative coordinate directions), but whether fewer suffice is unknown for $n \\geq 3$. The problem connects to convex geometry, discrete geometry, and optimization. Even the three-dimensional case ($n = 3$) remains open.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 198, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1187, "problem_number": "ALG-023", "title": "The Andrews-Curtis Conjecture", "statement": "Can every balanced presentation of the trivial group be transformed into a trivial presentation by a sequence of Nielsen transformations and conjugations?", "background": "Proposed in 1965 by James Andrews and Morton Curtis, this conjecture addresses the problem of simplifying group presentations. A balanced presentation has the same number of generators and relators. The question asks whether any such presentation of the trivial group can be reduced to the obvious trivial presentation through elementary operations (Nielsen transformations on relators and conjugating relators). Despite extensive computational searches, no counterexample has been found, but the general case remains unresolved. This problem connects combinatorial group theory, topology (via the Whitehead conjecture), and algorithmic complexity.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 412, "favorite_count": 28, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1188, "problem_number": "ALG-024", "title": "The Bounded Burnside Problem", "statement": "For which positive integers $m$ and $n$ is the free Burnside group $B(m,n)$ finite? In particular, is $B(2, 5)$ finite?", "background": "The Burnside problem, posed in 1902, asks whether a finitely generated group in which every element has finite order must itself be finite. The bounded version restricts to groups where all elements have order dividing a fixed $n$. Major breakthroughs came when Novikov and Adian (1968) proved $B(m,n)$ is infinite for odd $n \\geq 4381$ and $m \\geq 2$, and Zel'manov earned a Fields Medal (1994) for proving finiteness when $n$ is a prime power. The case $B(2,5)$ remains a famous open problem. The group $B(2,3)$ is known to be finite with 27 elements.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 687, "favorite_count": 52, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1189, "problem_number": "ALG-025", "title": "The Guralnick-Thompson Conjecture", "statement": "What are the composition factors of finite groups appearing in genus-0 systems?", "background": "This conjecture, proposed by Robert Guralnick and John Thompson, concerns the classification of finite groups that can act on Riemann surfaces of genus 0. The conjecture provides a list of simple groups that can appear as composition factors of such groups. The problem connects group theory with algebraic geometry and the theory of automorphisms of Riemann surfaces. Genus-0 systems are particularly important in the classification of finite simple groups and their actions on low-genus surfaces.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 298, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1190, "problem_number": "ALG-026", "title": "The Herzog-Schönheim Conjecture", "statement": "If a finite system of left cosets of subgroups of a group $G$ partitions $G$, must some two subgroups have the same index?", "background": "Proposed independently by Marcel Herzog and Jochanan Schönheim in 1974, this conjecture states that if finitely many left cosets of subgroups partition a group, then at least two of the subgroups must have the same finite index. This problem arises naturally in the study of group coverings and has connections to number theory through systems of covering congruences. Despite much research, the conjecture remains open in general, though it has been verified for various special cases including abelian groups and certain classes of finite groups.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 321, "favorite_count": 22, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1191, "problem_number": "ALG-027", "title": "The Inverse Galois Problem", "statement": "Is every finite group the Galois group of some Galois extension of $\\mathbb{Q}$?", "background": "The inverse Galois problem is one of the central open problems in Galois theory. While classical Galois theory establishes a correspondence between field extensions and groups, the inverse problem asks whether every finite group can be realized as the Galois group of an extension of the rational numbers. The problem was implicit in work of Hilbert and has been explicitly studied since the late 19th century. It has been solved affirmatively for many classes of groups (symmetric groups, alternating groups, many sporadic simple groups), but the general case remains open. The problem connects algebra, number theory, and algebraic geometry through its connection to dessins d'enfants and modular curves.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 892, "favorite_count": 67, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1192, "problem_number": "ALG-028", "title": "The Isomorphism Problem for Coxeter Groups", "statement": "Is there an algorithm to determine whether two Coxeter groups given by presentations are isomorphic?", "background": "Coxeter groups are fundamental objects in geometric group theory, generated by reflections with certain relations. They include the symmetry groups of regular polytopes and tessellations. The isomorphism problem asks whether there exists an algorithmic procedure to decide if two Coxeter groups, given by their Coxeter diagrams or presentations, are isomorphic. While the problem is solved for finite and affine Coxeter groups, the general case for arbitrary Coxeter groups remains open. This problem is related to the broader isomorphism problem for groups and has applications in geometry and topology.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 367, "favorite_count": 25, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1193, "problem_number": "ALG-029", "title": "Infinitude of Leinster Groups", "statement": "Are there infinitely many Leinster groups?", "background": "A Leinster group is a finite group whose order equals the sum of the orders of its proper normal subgroups. Named after Tom Leinster who studied them in 1996, only two examples are currently known: the cyclic group of order 6 and a group of order 12. The question of whether infinitely many such groups exist remains open. This problem connects group theory with number theory through the properties of divisors and has connections to the study of perfect numbers (where the sum of proper divisors equals the number itself).", "difficulty_level_id": 3, "status": "open", "category_id": 4, "view_count": 245, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1194, "problem_number": "ALG-030", "title": "Existence of Generalized Moonshine", "statement": "Does generalized moonshine exist for all elements of the Monster group?", "background": "Monstrous moonshine, discovered in the 1970s and proven by Borcherds (Fields Medal 1998), reveals a surprising connection between the Monster group (the largest sporadic simple group) and modular functions. Generalized moonshine extends this to other elements of the Monster group, asking whether similar connections exist for all group elements. Conway and Norton conjectured explicit relationships, and significant progress has been made, but the complete generalized moonshine remains unproven. This connects finite groups, modular forms, string theory, and vertex operator algebras in profound ways.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 543, "favorite_count": 41, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1195, "problem_number": "ALG-031", "title": "Finiteness of Finitely Presented Periodic Groups", "statement": "Is every finitely presented periodic group finite?", "background": "A periodic group (or torsion group) is one in which every element has finite order. The question of whether a finitely presented periodic group must be finite was a major open problem for much of the 20th century. The restricted Burnside problem, solved by Zel'manov, showed that finitely generated groups where all elements have bounded order must be finite. However, the general case without the bounded exponent assumption remains open. This problem connects group theory, geometric group theory, and algorithmic questions about group presentations.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 456, "favorite_count": 33, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1196, "problem_number": "ALG-032", "title": "The Surjunctivity Conjecture", "statement": "Is every group surjunctive?", "background": "A group is surjunctive if every injective cellular automaton over that group is also surjective. Equivalently, every injective endomorphism of the shift space is surjective. This property was introduced by Gottschalk in 1973 and connects symbolic dynamics, cellular automata theory, and group theory. Gromov and Weiss proved that all sofic groups are surjunctive, and since all amenable groups are sofic, this includes a large class. However, the general question of whether all groups are surjunctive remains open and is equivalent to asking whether all groups are sofic.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 389, "favorite_count": 27, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1197, "problem_number": "ALG-033", "title": "The Sofic Groups Conjecture", "statement": "Is every discrete countable group sofic?", "background": "A group is sofic if it can be approximated by finite symmetric groups in a precise sense. The concept was introduced by Gromov and Weiss around 1999 and has become central in modern group theory. All known groups are sofic: amenable groups, residually finite groups, linear groups, and many others. The soficity of all groups would have profound consequences for many conjectures in group theory, operator algebras, and ergodic theory. Notable implications include Connes' embedding conjecture (now known to be false via quantum complexity theory) and Gottschalk's surjunctivity conjecture. Despite the breadth of known sofic groups, the general question remains one of the deepest open problems in infinite group theory.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 612, "favorite_count": 48, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1198, "problem_number": "ALG-034", "title": "Arthur's Conjectures", "statement": "What is the structure of the discrete spectrum of automorphic forms on reductive groups?", "background": "Proposed by James Arthur in the 1980s, these conjectures describe the decomposition of the space of automorphic forms into irreducible representations. They provide a framework for understanding the discrete spectrum in terms of endoscopic groups and Arthur packets. The conjectures connect representation theory, harmonic analysis, and number theory, generalizing results of Langlands. Major progress has been made, including Arthur's proof for classical groups (2013), but the full program for all reductive groups remains incomplete. These conjectures are central to the Langlands program and have applications to trace formulas and L-functions.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 478, "favorite_count": 35, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1199, "problem_number": "ALG-035", "title": "Dade's Conjecture", "statement": "Is there a relationship between the numbers of irreducible characters in blocks of a finite group and its local subgroups?", "background": "Proposed by Everett Dade in 1992, this conjecture concerns the modular representation theory of finite groups. It relates the number of irreducible characters of a given defect in a block of a finite group to corresponding numbers in blocks of certain local subgroups (normalizers of p-subgroups). The conjecture is part of a broader program to reduce questions about representations of finite groups to questions about p-groups and their normalizers. It has been verified for many classes of groups but remains open in general. Dade's conjecture refines earlier conjectures by Alperin and McKay.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 312, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1200, "problem_number": "ALG-036", "title": "The Demazure Conjecture", "statement": "Can representations of semisimple algebraic groups be characterized over the integers?", "background": "Proposed by Michel Demazure in the 1970s, this conjecture concerns the existence of certain integral structures on representations of algebraic groups. It asks whether irreducible representations of semisimple algebraic groups over fields of positive characteristic can be deformed to characteristic zero while preserving integrality properties. The conjecture has applications to geometric representation theory and the theory of quantum groups. Partial results have been obtained for special cases, but the general conjecture remains open. The problem connects algebraic groups, representation theory, and arithmetic geometry.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 289, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1203, "problem_number": "GEO-012", "title": "The Spherical Bernstein Problem", "statement": "What is the classification of complete minimal hypersurfaces in spheres of all dimensions?", "background": "This is a generalization of Bernstein's problem (solved by 1968) which asked whether the only minimal graph over all of Euclidean space is a hyperplane. The spherical version asks for the classification of complete minimal hypersurfaces in the sphere $S^{n+1}$. While progress has been made in specific dimensions, a complete classification for all dimensions remains open. The problem connects differential geometry, minimal surface theory, and geometric analysis, with applications to general relativity and materials science.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 387, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1204, "problem_number": "GEO-013", "title": "The Carathéodory Conjecture", "statement": "Does every convex, closed, twice-differentiable surface in $\\mathbb{R}^3$ have at least two umbilical points?", "background": "Proposed by Constantin Carathéodory in the 1920s, this conjecture concerns umbilical points on convex surfaces—points where the principal curvatures are equal. The conjecture states that any smooth closed convex surface in 3-dimensional Euclidean space must have at least two such points. A sphere has infinitely many umbilical points (every point is umbilical), while an ellipsoid generically has exactly 4. The conjecture has been proven for surfaces of revolution and certain other special cases, but remains open in general. It connects differential geometry, topology (via the Poincaré-Hopf theorem), and dynamical systems.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 456, "favorite_count": 31, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1205, "problem_number": "GEO-014", "title": "The Cartan-Hadamard Conjecture", "statement": "Does the isoperimetric inequality hold for Cartan-Hadamard manifolds?", "background": "The classical isoperimetric inequality states that among all regions of fixed volume in Euclidean space, a ball has the smallest surface area. The Cartan-Hadamard conjecture asks whether this extends to Cartan-Hadamard manifolds—complete, simply connected Riemannian manifolds of nonpositive sectional curvature. The conjecture has been proven in dimensions 2, 3, and 4, and for many special classes of manifolds, but remains open in higher dimensions. This problem is central to geometric analysis and has connections to optimal transport theory, general relativity, and the study of black hole thermodynamics.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 523, "favorite_count": 39, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1206, "problem_number": "GEO-015", "title": "Chern's Affine Conjecture", "statement": "Does the Euler characteristic of a compact affine manifold vanish?", "background": "Proposed by Shiing-Shen Chern, this conjecture states that every closed affine manifold (a manifold with an atlas whose transition functions are affine transformations) has Euler characteristic zero. An affine structure is stronger than a smooth structure but weaker than a Riemannian structure. The conjecture has been verified for many classes of affine manifolds, and recent work has made substantial progress, but a complete proof remains elusive. The problem connects differential geometry, topology, and the theory of geometric structures on manifolds.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 398, "favorite_count": 27, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1207, "problem_number": "GEO-016", "title": "Chern's Conjecture for Hypersurfaces in Spheres", "statement": "What minimal hypersurfaces in spheres have constant mean curvature?", "background": "This is actually a family of related conjectures proposed by Shiing-Shen Chern concerning the classification of minimal and constant mean curvature hypersurfaces embedded in spheres. One version asks whether the only minimal hypersurface in $S^{n+1}$ with constant scalar curvature is the totally geodesic $S^n$. These conjectures connect minimal surface theory, the study of isoparametric hypersurfaces, and geometric analysis. Partial results have been obtained, but the general conjectures remain open.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 367, "favorite_count": 23, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1208, "problem_number": "GEO-017", "title": "The Closed Curve Problem", "statement": "What are necessary and sufficient conditions for an integral curve defined by two periodic functions to be closed?", "background": "This problem asks for explicit, computable conditions to determine when a curve defined parametrically by integrating two periodic functions with the same period will close up. The question arises naturally in dynamical systems, Hamiltonian mechanics, and the study of periodic orbits. While special cases are understood, general necessary and sufficient conditions that can be readily checked remain unknown. The problem connects analysis, differential geometry, and dynamical systems theory.", "difficulty_level_id": 3, "status": "open", "category_id": 6, "view_count": 289, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1209, "problem_number": "GEO-018", "title": "The Filling Area Conjecture", "statement": "Does a hemisphere have minimum area among shortcut-free surfaces with a given boundary length?", "background": "This conjecture in systolic geometry states that among all surfaces in Euclidean space whose boundary is a closed curve of given length and which contain no shortcuts (the surface distance between boundary points equals the Euclidean distance), the hemisphere has minimal area. The problem was proposed by Gromov and connects differential geometry, geometric measure theory, and the calculus of variations. It has applications to the study of minimal surfaces and optimal shapes in physics and materials science.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 334, "favorite_count": 22, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1210, "problem_number": "GEO-019", "title": "The Hopf Conjectures", "statement": "What is the relationship between curvature and Euler characteristic for even-dimensional Riemannian manifolds?", "background": "Heinz Hopf proposed several conjectures relating the sign of sectional curvature to the Euler characteristic and other topological invariants of closed Riemannian manifolds. The most famous asks whether a closed even-dimensional manifold with positive (or negative) sectional curvature must have positive Euler characteristic. The conjectures have been resolved in dimension 2 (Gauss-Bonnet) and partially in dimension 4, but remain open in higher dimensions. These problems are central to understanding the interplay between curvature and topology in Riemannian geometry.", "difficulty_level_id": 5, "status": "open", "category_id": 6, "view_count": 567, "favorite_count": 43, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1211, "problem_number": "GEO-020", "title": "The Osserman Conjecture", "statement": "Is every Osserman manifold either flat or locally isometric to a rank-one symmetric space?", "background": "An Osserman manifold is a Riemannian manifold where the eigenvalues of the Jacobi operator are constant on the unit sphere bundle at each point. Robert Osserman conjectured that such manifolds must be either flat or locally isometric to a rank-one symmetric space (spheres, projective spaces, or hyperbolic spaces). The conjecture has been proven in dimensions up to 4 and for many special cases, but remains open in higher dimensions. The problem connects differential geometry, spectral theory, and the theory of symmetric spaces.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 412, "favorite_count": 28, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1212, "problem_number": "GEO-021", "title": "Yau's Conjecture on First Eigenvalues", "statement": "Is the first eigenvalue of the Laplace-Beltrami operator on a minimal hypersurface in $S^{n+1}$ equal to $n$?", "background": "Proposed by Shing-Tung Yau, this conjecture states that for any closed embedded minimal hypersurface in the $(n+1)$-dimensional sphere $S^{n+1}$, the first nonzero eigenvalue of the Laplace-Beltrami operator equals $n$. This would provide a sharp spectral characterization of minimal hypersurfaces in spheres. The conjecture has been verified for several important cases including geodesic spheres, Clifford tori, and certain other symmetric examples. The problem connects spectral geometry, minimal surface theory, and PDEs on manifolds.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 478, "favorite_count": 34, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1213, "problem_number": "GEO-022", "title": "The Hadwiger Covering Conjecture", "statement": "Can every $n$-dimensional convex body be covered by at most $2^n$ smaller homothetic copies?", "background": "Proposed by Hugo Hadwiger in 1957, this conjecture states that any $n$-dimensional convex body can be covered by at most $2^n$ positive homothetic (scaled and translated) copies of itself with smaller ratio. The conjecture is known to be true for $n = 1$ (trivial) and $n = 2$ (proven), but remains open for $n \\geq 3$. The problem connects discrete geometry, convex geometry, and combinatorics. It is related to the illumination problem and has connections to coding theory and sphere packing.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 523, "favorite_count": 38, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1214, "problem_number": "GEO-023", "title": "The Happy Ending Problem", "statement": "What is the minimum number of points in the plane needed to guarantee a convex $n$-gon?", "background": "The Happy Ending problem, named by Paul Erdős because it led to the marriage of Esther Klein and George Szekeres, asks for $g(n)$—the smallest number such that any set of $g(n)$ points in general position contains $n$ points forming a convex $n$-gon. It's known that $2^{n-2} + 1 \\leq g(n) \\leq \\binom{2n-4}{n-2} + 1$. The exact value is known only for $n \\leq 6$. Erdős offered $500 for a proof that $g(n) = 2^{n-2} + 1$. The problem is central to combinatorial geometry and Ramsey theory.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 612, "favorite_count": 47, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1215, "problem_number": "GEO-024", "title": "The Heilbronn Triangle Problem", "statement": "What is the largest minimum area of a triangle determined by $n$ points in a unit square?", "background": "Proposed by Hans Heilbronn in 1908, this problem asks how to place $n$ points in a unit square to maximize the smallest area of any triangle they determine. Heilbronn originally conjectured the maximum was $O(1/n^2)$, but this was disproven—the actual order is between $\\Omega(\\log n / n^2)$ and $O(1/n^{8/7-\\epsilon})$. Finding the exact asymptotic remains open. The problem connects discrete geometry, extremal combinatorics, and has applications to numerical integration and computational geometry.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 445, "favorite_count": 31, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1216, "problem_number": "GEO-025", "title": "Kalai's $3^d$ Conjecture", "statement": "Does every centrally symmetric $d$-dimensional polytope have at least $3^d$ faces?", "background": "Proposed by Gil Kalai in 1989, this conjecture states that any centrally symmetric convex polytope in $d$ dimensions must have at least $3^d$ faces (including the polytope itself and the empty set). The bound is tight, achieved by the $d$-dimensional cube which has exactly $3^d$ faces. The conjecture has been verified for $d \\leq 4$ and for various special classes of polytopes. The problem connects combinatorics, convex geometry, and polytope theory, with applications to optimization and computational geometry.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 378, "favorite_count": 26, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1217, "problem_number": "GEO-026", "title": "The Unit Distance Problem", "statement": "What is the maximum number of unit distances determined by $n$ points in the plane?", "background": "This problem, posed by Erdős in 1946, asks for the maximum number of pairs of points at distance exactly 1 in a set of $n$ points in the Euclidean plane. The best known construction gives $\\Omega(n^{4/3})$ unit distances, while the best upper bound is $O(n^{4/3})$. Determining the exact asymptotic (and whether the exponent is exactly $4/3$) remains open. The problem connects extremal combinatorics, incidence geometry, and has applications to facility location and wireless network design. Erdős offered prizes for progress on this problem.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 567, "favorite_count": 42, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1219, "problem_number": "GEO-028", "title": "Ehrhart's Volume Conjecture", "statement": "Does a convex body in $\\mathbb{R}^n$ with one interior lattice point at its center of mass have volume at most $(n+1)^n/n!$?", "background": "Proposed by Eugène Ehrhart, this conjecture concerns lattice polytopes—convex bodies whose vertices have integer coordinates. It states that if a convex body in $n$ dimensions contains exactly one lattice point in its interior (which is its center of mass), then its volume cannot exceed $(n+1)^n/n!$, the volume of a regular simplex. The conjecture has been verified for $n \\leq 3$ and for many special cases. The problem connects discrete geometry, convex geometry, and number theory, with applications to integer programming and combinatorial optimization.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 389, "favorite_count": 27, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1220, "problem_number": "ALG-039", "title": "The Cherlin-Zilber Conjecture", "statement": "Is every simple group with a stable first-order theory an algebraic group over an algebraically closed field?", "background": "Proposed by Gregory Cherlin and Boris Zilber in the 1970s, this conjecture connects model theory and group theory. It states that any infinite simple group whose first-order theory is stable must be isomorphic to a simple algebraic group defined over an algebraically closed field. The conjecture has been verified for many classes of groups and represents a deep connection between logic and algebra. It generalizes the classification of finite simple groups to model-theoretic contexts and has implications for the structure theory of stable groups.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 412, "favorite_count": 29, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1221, "problem_number": "ALG-040", "title": "The Generalized Star Height Problem", "statement": "Can all regular languages be expressed with generalized regular expressions of bounded star height?", "background": "This problem in formal language theory asks whether there exists a uniform bound on the nesting depth of Kleene star operations needed to express any regular language using generalized regular expressions (which allow complementation). While the ordinary star height problem (without complementation) was solved—showing unbounded star height is necessary—the generalized version remains open. The problem connects automata theory, formal languages, and computational complexity, with applications to pattern matching and compiler design.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 334, "favorite_count": 23, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1222, "problem_number": "NT-031", "title": "Hilbert's Tenth Problem for Number Fields", "statement": "For which number fields is there an algorithm to determine solvability of Diophantine equations?", "background": "Hilbert's tenth problem asked for an algorithm to determine whether a Diophantine equation has integer solutions. Matiyasevich (building on work by Davis, Putnam, and Robinson) proved in 1970 that no such algorithm exists for the integers. The problem remains open for other rings, particularly number fields (finite extensions of the rationals). It has been solved negatively for some number fields and positively for others, but the general characterization is unknown. This connects logic, number theory, and computability theory.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 523, "favorite_count": 39, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1224, "problem_number": "GEO-029", "title": "Borsuk's Conjecture", "statement": "Can every bounded set in $\\mathbb{R}^n$ be partitioned into $n+1$ sets of smaller diameter?", "background": "Proposed by Karol Borsuk in 1933, this conjecture asks whether every bounded set in $n$-dimensional Euclidean space can be partitioned into $n+1$ parts, each with diameter strictly smaller than the original set. The conjecture held for dimensions up to 3 until 1993, when Kahn and Kalai found a counterexample in dimension 1325. The smallest dimension for which the conjecture fails remains unknown (known to fail for $n \\geq 64$). This problem connects geometric combinatorics, high-dimensional geometry, and has inspired research into diameter-reducing partitions.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 523, "favorite_count": 39, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1225, "problem_number": "GEO-030", "title": "The Kissing Number Problem", "statement": "What is the maximum number of non-overlapping unit spheres that can touch a central unit sphere in $n$ dimensions?", "background": "The kissing number $\\tau_n$ is the maximum number of non-overlapping unit spheres that can simultaneously touch a central unit sphere in $n$-dimensional Euclidean space. Known exactly only for dimensions 1, 2, 3, 4, 8, and 24, this problem has connections to sphere packing, coding theory, and lattice theory. The dimensions 8 and 24 are special due to exceptional lattices (E8 and Leech lattice). Determining kissing numbers in other dimensions, particularly dimensions 5, 6, 7, and general high dimensions, remains a major open problem in discrete geometry.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 612, "favorite_count": 46, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1226, "problem_number": "GEO-031", "title": "Ulam's Packing Conjecture", "statement": "Is the sphere the worst-packing convex solid?", "background": "Proposed by Stanisław Ulam, this conjecture asks which three-dimensional convex body has the smallest packing density. Ulam conjectured that the sphere is the worst-packing convex solid, meaning that among all convex bodies in 3D, spheres have the smallest proportion of space filled when packed. While the sphere packing problem (densest packing) was solved by Hales (2005), the worst-packing problem remains open. The conjecture connects packing theory, convex geometry, and optimization, with potential applications to materials science and crystallography.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 445, "favorite_count": 32, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1227, "problem_number": "GEO-032", "title": "Sphere Packing in High Dimensions", "statement": "What is the densest packing of unit spheres in dimensions other than 1, 2, 3, 8, and 24?", "background": "The sphere packing problem asks for the densest arrangement of non-overlapping unit spheres in $n$-dimensional Euclidean space. Solved for dimensions 1 and 2 (trivial), dimension 3 by Hales (1998, computer-assisted proof), dimension 8 by Viazovska (2016), and dimension 24 by Cohn et al. (2016), the problem remains open for all other dimensions. Understanding the asymptotic behavior as $n \\to \\infty$ is also open. This connects coding theory, lattice theory, and has applications to error-correcting codes and wireless communications.", "difficulty_level_id": 5, "status": "open", "category_id": 6, "view_count": 734, "favorite_count": 58, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1231, "problem_number": "COMB-010", "title": "The Cap Set Problem", "statement": "What is the maximum size of a cap set in $\\mathbb{F}_3^n$?", "background": "A cap set is a subset of the $n$-dimensional vector space over the three-element field with no three elements in arithmetic progression (analogous to the card game SET). The problem asks for the maximum size of such a set as a function of $n$. In 2016, Ellenberg and Gijswijt proved an upper bound of $O(2.756^n)$, dramatically improving previous bounds and resolving the longstanding question of whether cap sets can have exponential size $3^{cn}$ for $c > 0$. However, the exact maximum size and optimal constant remain open. This connects additive combinatorics, algebraic combinatorics, and theoretical computer science.", "difficulty_level_id": 4, "status": "open", "category_id": 2, "view_count": 523, "favorite_count": 40, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1235, "problem_number": "COMB-012", "title": "The Sunflower Conjecture", "statement": "Does every family of at least $c^k k!$ sets of size $k$ contain a sunflower of size 3, for some absolute constant $c$?", "background": "Proposed by Erdős and Rado in 1960, a sunflower (or $\\Delta$-system) is a collection of sets where every pair shares the same common intersection. The conjecture asks whether the exponential bound $c^k k!$ suffices to guarantee a sunflower of any fixed size. The best known bound is super-exponential. In 2019, Alweiss et al. made breakthrough progress by improving the bound to $O((\\log k)^k k!)$, but reaching the conjectured bound remains open. This problem is central to extremal combinatorics and has applications to circuit complexity and communication complexity.", "difficulty_level_id": 5, "status": "open", "category_id": 2, "view_count": 612, "favorite_count": 48, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1236, "problem_number": "COMB-013", "title": "Ramsey Number $R(5,5)$", "statement": "What is the exact value of the Ramsey number $R(5,5)$?", "background": "Ramsey numbers quantify the size at which complete disorder becomes impossible. $R(5,5)$ is the minimum number of vertices such that any two-coloring of the edges of the complete graph contains either a red $K_5$ or a blue $K_5$. It is known that $43 \\leq R(5,5) \\leq 48$, but the exact value remains unknown despite over 50 years of effort. This is perhaps the most famous open Ramsey number. Erdős famously suggested that finding $R(6,6)$ would require astronomical resources, but $R(5,5)$ seems tantalizingly within reach. The problem connects combinatorics, graph theory, and computational mathematics.", "difficulty_level_id": 4, "status": "open", "category_id": 2, "view_count": 823, "favorite_count": 67, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1239, "problem_number": "NT-032", "title": "Gauss Circle Problem", "statement": "How far can the number of lattice points in a circle centered at the origin deviate from the area of the circle?", "background": "The Gauss circle problem asks for the tightest bound on the error term in counting integer lattice points $(m,n)$ inside a circle of radius $r$ centered at the origin. The number of such points is $\\pi r^2 + E(r)$ where $E(r)$ is the error. It is known that $E(r) = O(r^{2/3})$ and $E(r) = \\Omega(r^{1/2} \\log r)$, but the exact growth rate remains unknown. Hardy conjectured $E(r) = O(r^{1/2+\\varepsilon})$ for any $\\varepsilon > 0$. This connects analytic number theory, lattice point enumeration, and has applications to physics and crystallography.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 478, "favorite_count": 35, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1240, "problem_number": "NT-033", "title": "Grimm's Conjecture", "statement": "Can each element of a set of consecutive composite numbers be assigned a distinct prime divisor?", "background": "Proposed by C. A. Grimm in 1969, this conjecture states that if we have $k$ consecutive composite numbers, then there exist $k$ distinct primes each dividing one of these numbers. For example, the consecutive composites $24, 25, 26, 27, 28$ have distinct prime divisors $3, 5, 13, 7, 2$ respectively. While verified computationally for large ranges, the general proof remains elusive. The conjecture connects to prime gaps, divisibility properties, and the distribution of primes among consecutive integers.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 412, "favorite_count": 29, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1241, "problem_number": "NT-034", "title": "Hall's Conjecture", "statement": "For any $\\varepsilon > 0$, is there a constant $c(\\varepsilon)$ such that either $y^2 = x^3$ or $|y^2 - x^3| > c(\\varepsilon) x^{1/2-\\varepsilon}$?", "background": "Proposed by Marshall Hall Jr. in 1970, this conjecture provides a measure of how close a perfect square can be to a perfect cube without being equal. It strengthens earlier work on Diophantine approximation and relates to the ABC conjecture. The conjecture has been verified for many special cases but remains open in general. It connects algebraic number theory, Diophantine equations, and elliptic curves, with implications for understanding integer solutions to polynomial equations.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 445, "favorite_count": 33, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1242, "problem_number": "NT-035", "title": "Lehmer's Totient Problem", "statement": "If Euler's totient function $\\phi(n)$ divides $n-1$, must $n$ be prime?", "background": "Posed by D. H. Lehmer in 1932, this problem asks whether any composite number $n$ exists such that $\\phi(n)$ divides $n-1$, where $\\phi(n)$ counts integers up to $n$ coprime to $n$. For all primes $p$, we have $\\phi(p) = p-1$, so the divisibility holds. Lehmer conjectured no composite number has this property. It has been verified that any such composite must be odd, square-free, and have at least 7 prime factors, with the smallest exceeding $10^{20}$. This connects Euler's totient function, primality, and multiplicative number theory.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 523, "favorite_count": 41, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1243, "problem_number": "NT-036", "title": "Magic Square of Squares", "statement": "Does there exist a 3×3 magic square composed entirely of distinct perfect squares?", "background": "A magic square has the property that all rows, columns, and diagonals sum to the same value. While magic squares of integers are well understood, the question of whether a 3×3 magic square can be constructed using only distinct perfect squares has remained open for centuries. Martin LaBar proved in 1984 that no such square exists using rational squares, but the integer case remains unsolved. Partial results exist for 4×4 and larger squares. This connects number theory, Diophantine equations, and recreational mathematics.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 589, "favorite_count": 47, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1244, "problem_number": "NT-037", "title": "Mahler's 3/2 Problem", "statement": "Is there a real number $x$ such that the fractional parts of $x(3/2)^n$ are all less than $1/2$ for every positive integer $n$?", "background": "Proposed by Kurt Mahler in the 1960s, this problem concerns the distribution of the sequence $\\{x(3/2)^n\\}$ modulo 1, where $\\{y\\}$ denotes the fractional part of $y$. Mahler conjectured that no such $x$ exists. The problem relates to ergodic theory, uniform distribution, and Diophantine approximation. While various partial results have been obtained using techniques from dynamical systems and number theory, the general question remains open. It exemplifies deep questions about the behavior of geometric sequences under modular arithmetic.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 398, "favorite_count": 28, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1245, "problem_number": "NT-038", "title": "Newman's Conjecture", "statement": "Does the partition function satisfy any arbitrary congruence infinitely often?", "background": "Proposed by Morris Newman, this conjecture concerns the partition function $p(n)$, which counts the number of ways to write $n$ as a sum of positive integers. Newman conjectured that for any integers $a$ and $m$ with $\\gcd(a,m) = 1$, there are infinitely many $n$ such that $p(n) \\equiv a \\pmod{m}$. This would imply the partition function takes all possible residue classes modulo any integer infinitely often. The conjecture connects partition theory, modular forms, and has implications for understanding the arithmetic properties of partitions.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 367, "favorite_count": 26, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1246, "problem_number": "NT-039", "title": "Scholz Conjecture", "statement": "Is the shortest addition chain for $2^n - 1$ at most $n - 1$ plus the length of the shortest addition chain for $n$?", "background": "An addition chain for $m$ is a sequence $1 = a_0 < a_1 < \\cdots < a_r = m$ where each $a_i$ (for $i > 0$) is the sum of two earlier terms. Scholz conjectured in 1937 that $\\ell(2^n-1) \\leq n-1+\\ell(n)$ where $\\ell(m)$ denotes the minimum length of an addition chain for $m$. This has applications to efficient exponentiation algorithms in computer science and cryptography. While verified for many values and various special cases proven, the general conjecture remains open. It connects additive number theory, combinatorial optimization, and computational complexity.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 412, "favorite_count": 30, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1248, "problem_number": "NT-041", "title": "Infinitely Many Perfect Numbers", "statement": "Are there infinitely many perfect numbers?", "background": "A perfect number equals the sum of its proper divisors (divisors excluding itself). Examples include 6 = 1+2+3 and 28 = 1+2+4+7+14. Euclid proved that if $2^p - 1$ is prime (a Mersenne prime), then $2^{p-1}(2^p-1)$ is perfect. All known perfect numbers have this form and are even. Whether infinitely many exist depends on whether there are infinitely many Mersenne primes, itself an open question. The problem connects prime number theory, divisor functions, and has fascinated mathematicians for over 2000 years.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 678, "favorite_count": 54, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1250, "problem_number": "NT-043", "title": "Quasiperfect Numbers", "statement": "Do quasiperfect numbers exist?", "background": "A quasiperfect number is a natural number $n$ such that the sum of its divisors equals $2n + 1$ (one more than twice the number). No quasiperfect number has ever been found. It has been proven that if one exists, it must be an odd square number greater than $10^{35}$, and have at least seven distinct prime factors. The search for quasiperfect numbers connects divisor theory, multiplicative number theory, and computational number theory. Their existence or non-existence would provide insights into the structure of highly composite numbers.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 398, "favorite_count": 28, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1251, "problem_number": "NT-044", "title": "Almost Perfect Numbers Beyond Powers of 2", "statement": "Do any almost perfect numbers exist that are not powers of 2?", "background": "An almost perfect number $n$ has the sum of its proper divisors equal to $n - 1$. All powers of 2 are almost perfect, since the divisors of $2^k$ are $1, 2, 4, \\ldots, 2^{k-1}$ which sum to $2^k - 1$. It remains unknown whether any odd almost perfect number exists, or any even almost perfect number that is not a power of 2. The problem connects perfect numbers, divisor functions, and the structure of highly specific arithmetic sequences.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 356, "favorite_count": 25, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1252, "problem_number": "NT-045", "title": "The Number of Idoneal Numbers", "statement": "Are there exactly 65 idoneal numbers, or could there be 66 or 67?", "background": "Idoneal numbers (also called suitable or convenient numbers) are positive integers $D$ such that if $n = ax^2 + by^2$ with coprime $a,b$ is uniquely representable, then $n$ is a prime power or twice a prime power. Euler conjectured 65 such numbers exist, the largest being 1848. Weinberger proved in 1973 that at most one more exists, and if the generalized Riemann hypothesis is true, exactly 65 exist. This connects binary quadratic forms, class field theory, and the Riemann hypothesis. The resolution depends on deep questions in analytic number theory.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 334, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1253, "problem_number": "NT-046", "title": "Amicable Numbers of Opposite Parity", "statement": "Do any pairs of amicable numbers exist where one is odd and one is even?", "background": "Two numbers are amicable if each equals the sum of the proper divisors of the other. For example, 220 and 284 are amicable (both even). Over 12 million amicable pairs are known, all with matching parity (both even or both odd). It remains unknown whether a mixed-parity pair exists. Such a pair would require unusual divisor properties. The problem connects divisor sums, parity constraints, and the arithmetic structure of amicable pairs. All known odd amicable pairs have been found by Erdős and collaborators.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 389, "favorite_count": 27, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1254, "problem_number": "NT-047", "title": "Infinitely Many Amicable Pairs", "statement": "Are there infinitely many pairs of amicable numbers?", "background": "Amicable numbers are pairs where each number equals the sum of the other's proper divisors. While over 12 million pairs have been discovered, it remains unknown whether infinitely many exist. Thabit ibn Qurra (9th century) gave a formula generating some pairs, and Euler found many more. Various conjectures suggest their density, but no proof of infinitude exists. This contrasts with related questions like twin primes (conjectured infinite) and perfect numbers (infinitude depends on Mersenne primes). The problem connects multiplicative number theory and divisor sums.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 445, "favorite_count": 33, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1255, "problem_number": "NT-048", "title": "Infinitely Many Giuga Numbers", "statement": "Are there infinitely many Giuga numbers?", "background": "A Giuga number is a composite number $n$ such that $p$ divides $(n/p - 1)$ for every prime divisor $p$ of $n$. Equivalently, $\\sum_{p|n} (1/p) - 1/n$ is an integer. Only 15 Giuga numbers are known, the smallest being 30. Giuga conjectured that if $1 + \\sum_{i=1}^{n-1} i^{n-1} \\equiv 0 \\pmod{n}$ for composite $n$, then $n$ is a Giuga number. Whether infinitely many exist remains open. This connects primality testing, Carmichael numbers, and Fermat pseudoprimes.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 367, "favorite_count": 26, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1256, "problem_number": "NT-049", "title": "Lychrel Numbers in Base 10", "statement": "Do Lychrel numbers exist in base 10?", "background": "A Lychrel number is a natural number that never forms a palindrome through the iterative process of adding it to its reverse. For example, 89 is not Lychrel: 89 + 98 = 187, 187 + 781 = 968, 968 + 869 = 1837, 1837 + 7381 = 9218, 9218 + 8129 = 17347, 17347 + 74371 = 91718, 91718 + 81719 = 173437, 173437 + 734371 = 907808, 907808 + 808709 = 1716517, 1716517 + 7156171 = 8872688, which is a palindrome. The number 196 is the smallest candidate Lychrel number, having been tested to over 300 million iterations without producing a palindrome. No Lychrel number has been proven to exist.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 512, "favorite_count": 39, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1257, "problem_number": "NT-050", "title": "Odd Weird Numbers", "statement": "Do any odd weird numbers exist?", "background": "A weird number is a natural number that is abundant (the sum of its proper divisors exceeds the number) but not semiperfect (no subset of its divisors sums to the number). The smallest weird number is 70. All known weird numbers are even, and it has been conjectured that no odd weird numbers exist. If an odd weird number exists, it must be greater than $10^{21}$ and have at least 4 distinct prime factors. This connects abundant numbers, partition theory, and subset sum problems.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 378, "favorite_count": 27, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1258, "problem_number": "NT-051", "title": "Normality of Pi", "statement": "Is $\\pi$ a normal number in base 10?", "background": "A number is normal in base 10 if every digit 0-9 appears with equal frequency (1/10) in its decimal expansion, and more generally, every sequence of $k$ digits appears with frequency $1/10^k$. While the digits of $\\pi$ appear statistically random in computational tests extending to trillions of digits, no proof of normality exists. It is not even known whether every digit appears infinitely often in $\\pi$. Proving normality would require deep insights into the arithmetic nature of $\\pi$ and transcendental number theory.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 823, "favorite_count": 68, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1259, "problem_number": "NT-052", "title": "Normality of Irrational Algebraic Numbers", "statement": "Are all irrational algebraic numbers normal in every base?", "background": "An algebraic number is a root of a polynomial with integer coefficients. Normal numbers have every digit sequence appear with the expected frequency in their base expansions. It is conjectured that all irrational algebraic numbers like $\\sqrt{2}$ are normal in every integer base, but not a single irrational algebraic number has been proven normal in any base. This represents a fundamental gap in our understanding of the decimal expansions of algebraic numbers. The question connects algebraic number theory, Diophantine approximation, and the theory of normal numbers.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 567, "favorite_count": 45, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1260, "problem_number": "NT-053", "title": "Is 10 a Solitary Number?", "statement": "Is 10 a solitary number (no other number shares its abundancy index)?", "background": "The abundancy index of $n$ is $\\sigma(n)/n$ where $\\sigma(n)$ is the sum of divisors of $n$. A number is solitary if no other number has the same abundancy index. For 10, we have $\\sigma(10) = 1+2+5+10 = 18$, giving abundancy $18/10 = 9/5$. It remains unknown whether any other number has abundancy $9/5$. Numbers in amicable pairs and sociable numbers are not solitary. Many numbers have been proven non-solitary, but 10 resists classification. This connects divisor functions, Diophantine equations, and the classification of multiplicative structures.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 334, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1262, "problem_number": "NT-055", "title": "Erdős Conjecture on Arithmetic Progressions", "statement": "If the sum of reciprocals of a set of positive integers diverges, does the set contain arbitrarily long arithmetic progressions?", "background": "Erdős conjectured that if $A \\subseteq \\mathbb{N}$ and $\\sum_{a \\in A} 1/a = \\infty$, then $A$ contains arithmetic progressions of arbitrary length. This strengthens Szemerédi's theorem, which only requires positive density. The conjecture remains open even for progressions of length 3. In 2020, Bloom and Sisask made major progress by proving that if $\\sum_{a \\in A, a \\leq N} 1/a \\geq (\\log N)^{c \\log \\log \\log N}$ for some $c$, then $A$ contains a 3-term arithmetic progression. This connects additive combinatorics, harmonic analysis, and analytic number theory.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 534, "favorite_count": 42, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1263, "problem_number": "NT-056", "title": "Erdős-Turán Conjecture on Additive Bases", "statement": "If $B$ is an additive basis of order 2, must the representation function tend to infinity?", "background": "An additive basis of order 2 is a set $B$ such that every sufficiently large integer can be written as the sum of two elements of $B$. The representation function $r_B(n)$ counts the number of ways to write $n$ as $b_1 + b_2$ with $b_1, b_2 \\in B$. Erdős and Turán conjectured in 1941 that if $B$ is an additive basis of order 2, then $r_B(n)$ must tend to infinity. This has been proven for various special bases, but the general case remains open. The conjecture connects additive number theory, combinatorics, and the structure of thin bases.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 456, "favorite_count": 34, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1265, "problem_number": "NT-058", "title": "Lander-Parkin-Selfridge Conjecture", "statement": "If the sum of $m$ $k$-th powers equals the sum of $n$ $k$-th powers, must $m + n \\geq k$?", "background": "This conjecture generalizes Fermat's Last Theorem to sums of powers. It states that if $a_1^k + \\cdots + a_m^k = b_1^k + \\cdots + b_n^k$ with positive integers and the two sums are different, then $m + n \\geq k$. Euler conjectured the stronger statement that at least $k$ $k$-th powers are needed, but this was disproved: $27^5 + 84^5 + 110^5 + 133^5 = 144^5$ (counterexample with $k=5$, $m=4$, $n=1$). The weaker LPS conjecture remains open for $k \\geq 4$ and has implications for Diophantine equations and additive number theory.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 489, "favorite_count": 37, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1266, "problem_number": "NT-059", "title": "Lemoine's Conjecture", "statement": "Can every odd integer greater than 5 be expressed as the sum of an odd prime and an even semiprime?", "background": "Proposed by Émile Lemoine in 1894, this conjecture states that every odd number $n > 5$ can be written as $n = p + 2q$ where $p$ and $q$ are primes. An even semiprime is twice a prime. For example, $27 = 13 + 2(7)$, $31 = 19 + 2(6)$ is invalid since 6 isn't prime, but $31 = 5 + 2(13)$ works. This is weaker than Goldbach's conjecture (which implies Lemoine's). Verified computationally to very large numbers, but no proof exists. It connects prime distribution, additive representations, and the Goldbach problem.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 445, "favorite_count": 33, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1267, "problem_number": "NT-060", "title": "Recamán's Sequence Completeness", "statement": "Does every nonnegative integer appear in Recamán's sequence?", "background": "Recamán's sequence starts with $a_0 = 0$ and follows the rule: $a_n = a_{n-1} - n$ if that value is positive and not already in the sequence, otherwise $a_n = a_{n-1} + n$. This produces: 0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, ... Named after Colombian mathematician Bernardo Recamán Santos, this sequence has been computed to millions of terms, but it remains unknown whether every nonnegative integer appears. Some values appear very late or may never appear. This connects integer sequences, graph theory, and computational number theory.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 512, "favorite_count": 40, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1268, "problem_number": "NT-061", "title": "Skolem Problem", "statement": "Can an algorithm determine if a constant-recursive sequence contains a zero?", "background": "A constant-recursive sequence satisfies a linear recurrence with constant coefficients, like the Fibonacci sequence. The Skolem problem asks whether there exists an algorithm to determine if such a sequence ever equals zero. This is known to be decidable for sequences of order up to 4, but the general problem remains open. The Positivity Problem (whether all terms are positive) and Ultimate Positivity (whether terms are eventually all positive) are related variants. This connects computability theory, Diophantine approximation, and decidability in number theory.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 389, "favorite_count": 28, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1269, "problem_number": "NT-062", "title": "Waring's Problem: Exact Values", "statement": "What are the exact values of $g(k)$ and $G(k)$ for all $k$ in Waring's problem?", "background": "Waring's problem concerns representing integers as sums of $k$-th powers. Let $g(k)$ be the minimum number such that every positive integer can be written as a sum of at most $g(k)$ $k$-th powers, allowing any number of terms. Let $G(k)$ be the same but excluding a finite set of exceptions. We know $g(2)=4$ (Lagrange), $G(2)=4$, $g(3)=9$, $G(3)=4$, $g(4)=19$, $G(4)=16$. For general $k$, Hilbert proved $g(k)$ exists but exact values remain unknown for most $k$. This is a central problem in additive number theory.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 567, "favorite_count": 44, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1270, "problem_number": "NT-063", "title": "Density of Ulam Numbers", "statement": "Do the Ulam numbers have a positive density?", "background": "The Ulam numbers start with 1, 2, and each subsequent number is the smallest integer that can be expressed as the sum of two distinct earlier Ulam numbers in exactly one way: 1, 2, 3, 4, 6, 8, 11, 13, 16, 18, ... Named after Stanisław Ulam, these numbers appear to have density around 0.07, but whether the density exists and is positive remains unproven. Related questions about their growth rate and distribution connect to additive combinatorics, unique representation bases, and computational number theory.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 398, "favorite_count": 29, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1271, "problem_number": "NT-064", "title": "Class Number Problem", "statement": "Are there infinitely many real quadratic number fields with unique factorization?", "background": "A number field has unique factorization if every nonzero element factors uniquely into irreducibles. For real quadratic fields $\\mathbb{Q}(\\sqrt{d})$ with $d > 0$ square-free, unique factorization is equivalent to having class number 1. Gauss conjectured infinitely many such fields exist. While infinitely many imaginary quadratic fields (class number 1) were ruled out, the real case remains open. Computational evidence strongly supports the conjecture, but a proof eludes us. This connects algebraic number theory, class field theory, and the distribution of number fields.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 478, "favorite_count": 36, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1272, "problem_number": "NT-065", "title": "Hilbert's Twelfth Problem", "statement": "Can the Kronecker-Weber theorem on abelian extensions of $\\mathbb{Q}$ be extended to any base number field?", "background": "The Kronecker-Weber theorem states that every abelian extension of the rationals $\\mathbb{Q}$ is contained in a cyclotomic field (generated by roots of unity). Hilbert's 12th problem asks for an analogous explicit construction of abelian extensions of arbitrary number fields. For imaginary quadratic fields, complex multiplication provides a partial answer using elliptic curves and modular functions. For general number fields, the problem remains largely open despite over a century of work. This is fundamental to class field theory and arithmetic geometry.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 512, "favorite_count": 40, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1273, "problem_number": "NT-066", "title": "Leopoldt's Conjecture", "statement": "Does the $p$-adic regulator of an algebraic number field not vanish?", "background": "Leopoldt's conjecture, proposed in 1962, states that the $p$-adic regulator of an algebraic number field $K$ is nonzero for every prime $p$. The regulator measures the \"size\" of the unit group. The conjecture has been verified for abelian extensions of $\\mathbb{Q}$ and many other special cases, but remains open in general. It has deep connections to Iwasawa theory, $p$-adic L-functions, and the structure of class groups. A proof would have significant implications for understanding $p$-adic analytic properties of number fields.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 389, "favorite_count": 29, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1274, "problem_number": "NT-067", "title": "Lindelöf Hypothesis", "statement": "For all $\\varepsilon > 0$, does $\\zeta(1/2 + it) = o(t^\\varepsilon)$ as $t \\to \\infty$?", "background": "The Lindelöf hypothesis concerns the growth rate of the Riemann zeta function $\\zeta(s)$ on the critical line $\\text{Re}(s) = 1/2$. It states that for any $\\varepsilon > 0$, we have $|\\zeta(1/2 + it)| = o(t^\\varepsilon)$. This is weaker than the Riemann Hypothesis but still unproven. The best known bound is $O(t^{13/84+\\varepsilon})$ due to Bourgain (2022). The hypothesis has implications for the distribution of primes, zero-free regions of $\\zeta(s)$, and analytic number theory. It connects to moment problems and random matrix theory.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 545, "favorite_count": 43, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1275, "problem_number": "NT-068", "title": "Hilbert-Pólya Conjecture", "statement": "Do the nontrivial zeros of the Riemann zeta function correspond to eigenvalues of a self-adjoint operator?", "background": "The Hilbert-Pólya conjecture proposes a spectral interpretation of the Riemann Hypothesis: the nontrivial zeros of $\\zeta(s)$ at $1/2 + i\\gamma_n$ correspond to eigenvalues of some self-adjoint operator, with $\\gamma_n$ being the eigenvalues. This would imply RH since eigenvalues of self-adjoint operators are real. Connections to random matrix theory (Montgomery-Dyson) and quantum chaos support this idea. Finding such an operator remains elusive despite attempts involving quantum mechanics, trace formulas, and noncommutative geometry. This bridges analysis, spectral theory, and mathematical physics.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 623, "favorite_count": 51, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1276, "problem_number": "NT-069", "title": "Grand Riemann Hypothesis", "statement": "Do all automorphic L-functions have their nontrivial zeros on the critical line?", "background": "The Grand Riemann Hypothesis extends RH to all automorphic L-functions, a vast class including Dirichlet L-functions, Dedekind zeta functions, and L-functions of modular forms. It asserts that all nontrivial zeros lie on the critical line $\\text{Re}(s) = 1/2$. This would have profound consequences for prime distribution in arithmetic progressions, algebraic number theory, and the Langlands program. The GRH is considered one of the most important unifying conjectures in mathematics, generalizing many individual cases of the Riemann Hypothesis.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 712, "favorite_count": 59, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1277, "problem_number": "NT-070", "title": "Montgomery's Pair Correlation Conjecture", "statement": "Does the pair correlation function of Riemann zeta zeros match that of random Hermitian matrices?", "background": "Montgomery conjectured in 1973 that the statistical distribution of gaps between zeros of the Riemann zeta function matches the pair correlation of eigenvalues from the Gaussian Unitary Ensemble (GUE) of random matrix theory. This remarkable connection between number theory and quantum physics was discovered through numerical experiments and Dyson's insights. The conjecture has been partially verified but remains unproven. It suggests deep links between prime numbers, quantum chaos, and statistical mechanics, forming a cornerstone of the modern approach to understanding zeta zeros.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 567, "favorite_count": 46, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1278, "problem_number": "NT-071", "title": "Dirichlet's Divisor Problem", "statement": "What is the optimal exponent in the error term for the divisor summatory function?", "background": "Let $D(x) = \\sum_{n \\leq x} d(n)$ where $d(n)$ counts the divisors of $n$. Dirichlet proved $D(x) = x \\log x + (2\\gamma - 1)x + \\Delta(x)$ where $\\gamma$ is Euler's constant and $\\Delta(x)$ is the error. The problem asks for the infimum $\\theta$ such that $\\Delta(x) = O(x^\\theta)$. It is known that $1/4 \\leq \\theta < 131/416 \\approx 0.314903$. The Riemann Hypothesis would imply $\\theta \\leq 1/4 + \\varepsilon$ for any $\\varepsilon > 0$, but proving this is extremely difficult. This connects analytic number theory, the Riemann zeta function, and lattice point problems.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 445, "favorite_count": 34, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1279, "problem_number": "GEO-033", "title": "Erdős-Ulam Problem", "statement": "Is there a dense set of points in the plane with all pairwise distances rational?", "background": "Proposed by Paul Erdős and Stanisław Ulam, this problem asks whether there exists a dense subset of the Euclidean plane (dense in the usual topology) such that the distance between any two points is a rational number. While finite and countable dense sets with rational distances are known (like rational points on a circle), an everywhere-dense set remains undiscovered. The problem connects geometry, Diophantine equations, and the structure of rational points. It has implications for understanding constraints on rational distance sets.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 478, "favorite_count": 36, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1281, "problem_number": "NT-073", "title": "Four Exponentials Conjecture", "statement": "If $x_1, x_2$ are linearly independent over $\\mathbb{Q}$ and $y_1, y_2$ are linearly independent over $\\mathbb{Q}$, is at least one of $e^{x_1 y_1}, e^{x_1 y_2}, e^{x_2 y_1}, e^{x_2 y_2}$ transcendental?", "background": "This conjecture, a consequence of Schanuel's conjecture, asserts that under the stated conditions, at least one of the four exponentials must be transcendental. The six exponentials theorem (proven) states that if $x_1, x_2, x_3$ are $\\mathbb{Q}$-linearly independent and $y_1, y_2$ are $\\mathbb{Q}$-linearly independent, then among the six values $e^{x_i y_j}$, at least one is transcendental. The four exponentials conjecture would strengthen this. It connects exponential Diophantine equations, transcendence theory, and algebraic independence.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 445, "favorite_count": 34, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1282, "problem_number": "NT-074", "title": "Irrationality of Euler's Constant", "statement": "Is the Euler-Mascheroni constant $\\gamma$ irrational?", "background": "Euler's constant $\\gamma = \\lim_{n \\to \\infty} (1 + 1/2 + 1/3 + \\cdots + 1/n - \\ln n) \\approx 0.5772$ appears throughout mathematics but its arithmetic nature remains mysterious. It is not even known whether $\\gamma$ is irrational, let alone transcendental. While computational evidence suggests irrationality (verified to billions of digits), no proof exists. The problem has resisted attack for over 250 years. Progress would require new techniques in transcendental number theory and might illuminate the nature of other constants like $\\zeta(3)$.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 712, "favorite_count": 58, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1283, "problem_number": "NT-075", "title": "Transcendence of Apéry's Constant", "statement": "Is $\\zeta(3) = 1 + 1/8 + 1/27 + 1/64 + \\cdots$ transcendental?", "background": "Apéry's constant $\\zeta(3) \\approx 1.202$ is the value of the Riemann zeta function at 3. Roger Apéry proved its irrationality in 1978 using ingenious continued fraction methods, surprising the mathematical community. Whether $\\zeta(3)$ is transcendental remains unknown. More generally, the transcendence of $\\zeta(2k+1)$ for integer $k \\geq 1$ is open (except $\\zeta(1)$ which diverges). Rivoal (2000) proved infinitely many $\\zeta(2k+1)$ are irrational, and at least one of $\\zeta(5), \\zeta(7), \\zeta(9), \\zeta(11)$ is irrational, but transcendence is far harder.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 589, "favorite_count": 47, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1284, "problem_number": "NT-076", "title": "Littlewood Conjecture", "statement": "For any two real numbers $\\alpha, \\beta$, does $\\liminf_{n \\to \\infty} n \\|n\\alpha\\| \\|n\\beta\\| = 0$?", "background": "Proposed by John Edensor Littlewood around 1930, where $\\|x\\|$ denotes the distance from $x$ to the nearest integer. The conjecture asserts a simultaneous approximation property: for any pair of real numbers, infinitely many integers $n$ exist such that both $n\\alpha$ and $n\\beta$ are simultaneously close to integers, with the product of distances approaching zero. While verified for many cases (algebraic numbers, certain combinations), the general conjecture remains open. Einsiedler, Katok, and Lindenstrauss (2006) proved the set of counterexamples has Hausdorff dimension zero, suggesting counterexamples are rare if they exist.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 456, "favorite_count": 35, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1285, "problem_number": "NT-077", "title": "Integer Factorization in Polynomial Time", "statement": "Can integer factorization be solved in polynomial time on a classical computer?", "background": "The integer factorization problem asks: given a composite number $n$, find its prime factors. The best known classical algorithm (general number field sieve) runs in sub-exponential time $\\exp(O((\\ln n)^{1/3}(\\ln \\ln n)^{2/3}))$. Whether a polynomial-time classical algorithm exists is unknown and has profound implications for cryptography (RSA security relies on factorization hardness). Shor's algorithm solves factorization in polynomial time on quantum computers, but practical quantum computers don't yet exist. The problem connects computational complexity, cryptography, and number theory.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 734, "favorite_count": 61, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1286, "problem_number": "NT-078", "title": "Beal's Conjecture", "statement": "For $A^x + B^y = C^z$ with $x, y, z > 2$, must $A$, $B$, and $C$ share a common prime factor?", "background": "Proposed by banker and amateur mathematician Andrew Beal in 1993, this conjecture generalizes Fermat's Last Theorem. It asserts that if $A^x + B^y = C^z$ where $A, B, C, x, y, z$ are positive integers with $x, y, z > 2$, then $A$, $B$, and $C$ must have a common prime factor. For example, $3^3 + 6^3 = 3^5$ satisfies this since all share factor 3. Beal has offered a prize of $1 million for a proof or counterexample. The conjecture is equivalent to saying no solutions exist when $A$, $B$, $C$ are coprime. This connects Fermat's Last Theorem, the abc conjecture, and exponential Diophantine equations.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 712, "favorite_count": 59, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1289, "problem_number": "NT-081", "title": "Fermat-Catalan Conjecture", "statement": "Are there finitely many solutions to $a^m + b^n = c^k$ with coprime $a,b,c$ and $1/m + 1/n + 1/k < 1$?", "background": "This conjecture generalizes both Fermat's Last Theorem and the Catalan-Mersenne conjecture. It asserts that the equation $a^m + b^n = c^k$ has only finitely many solutions in coprime positive integers $a,b,c$ and integers $m,n,k \\geq 2$ satisfying $1/m + 1/n + 1/k < 1$. Ten solutions are known, including $1^m + 2^3 = 3^2$, $2^5 + 7^2 = 3^4$, and $17^3 + 2^{7\\cdot 13^3} = 71^2 \\cdot 13^3$. Beal's conjecture and the abc conjecture both imply Fermat-Catalan. The problem is central to exponential Diophantine equations.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 634, "favorite_count": 52, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1292, "problem_number": "NT-084", "title": "Bunyakovsky Conjecture", "statement": "Does an irreducible integer polynomial with no fixed prime divisor produce infinitely many primes?", "background": "Proposed by Viktor Bunyakovsky in 1857, this generalizes Dirichlet's theorem on primes in arithmetic progressions. It states that if polynomial $f(x)$ has integer coefficients, positive leading coefficient, is irreducible over integers, and has no common prime divisor of all its values $f(n)$ for positive integers $n$, then $f(x)$ represents infinitely many primes. This would imply infinitely many twin primes (using $f(x) = x$ and $g(x) = x+2$), Sophie Germain primes, and many other families. Despite being over 160 years old, it remains unproven except for linear polynomials (Dirichlet's theorem).", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 512, "favorite_count": 41, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1293, "problem_number": "NT-085", "title": "Dickson's Conjecture", "statement": "Do finitely many linear forms simultaneously take prime values infinitely often, barring congruence obstructions?", "background": "Proposed by Leonard Eugene Dickson in 1904, this generalizes Dirichlet's theorem and implies many prime conjectures. For linear forms $a_1 + b_1 n, \\ldots, a_k + b_k n$ with each $b_i \\geq 1$, if no congruence condition forces a composite, then infinitely many $n$ exist making all forms simultaneously prime. This would imply: twin primes, Sophie Germain primes, prime triplets, Goldbach's conjecture, and more. It strengthens Bunyakovsky and is a special case of Schinzel's Hypothesis H. No proof exists even for two forms, representing a fundamental gap in our understanding of simultaneous prime values.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 445, "favorite_count": 34, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1294, "problem_number": "NT-086", "title": "Brocard's Conjecture (Prime Gaps)", "statement": "Are there always at least 4 primes between consecutive squares of primes $p_n^2$ and $p_{n+1}^2$?", "background": "Proposed by Henri Brocard in 1904, this conjecture concerns the density of primes near perfect squares. For consecutive primes $p_n$ and $p_{n+1}$, Brocard conjectured there are always at least 4 primes in the interval $(p_n^2, p_{n+1}^2)$, except for the cases $(2^2, 3^2)$ which contains only one prime (5). Verified computationally to enormous values, but no proof exists. This is stronger than Legendre's conjecture (at least one prime between consecutive squares). It connects prime gaps, Bertrand's postulate generalizations, and the distribution of primes.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 398, "favorite_count": 29, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1295, "problem_number": "NT-087", "title": "Agoh-Giuga Conjecture", "statement": "Is $p$ prime if and only if $pB_{p-1} \\equiv -1 \\pmod{p}$ for the Bernoulli number $B_{p-1}$?", "background": "This conjecture combines work of Takashi Agoh (1990) and Giuseppe Giuga (1950), providing a primality criterion via Bernoulli numbers. Bernoulli numbers $B_n$ appear in number theory and analysis. The conjecture states: $p$ is prime iff $pB_{p-1} \\equiv -1 \\pmod{p}$. The forward direction is known (if $p$ prime, the congruence holds). The converse would give a new primality test. Related to Giuga numbers and Wolstenholme's theorem, this connects Bernoulli numbers, primality testing, and modular arithmetic in unexpected ways.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 334, "favorite_count": 25, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1296, "problem_number": "NT-088", "title": "Elliott-Halberstam Conjecture", "statement": "Do primes distribute uniformly in arithmetic progressions up to nearly $x$ (instead of $x^{1/2}$)?", "background": "Proposed in 1968, this strengthens the Bombieri-Vinogradov theorem about primes in arithmetic progressions. For most moduli $q < x^\\theta$, the primes are equidistributed among valid residue classes. Bombieri-Vinogradov proves this for $\\theta < 1/2$. Elliott-Halberstam conjectures it holds for any $\\theta < 1$. This would have dramatic consequences: it implies infinitely many bounded prime gaps exist (a weak form proven by Zhang 2013, then Polymath improved to gap 246). The full conjecture would likely yield bounded gaps near the twin prime level. It connects sieve methods, Goldston-Pintz-Yıldırım techniques, and multiplicative functions.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 412, "favorite_count": 32, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1297, "problem_number": "ALG-001", "title": "Birch–Tate Conjecture", "statement": "Is there a relation between the order of the center of the Steinberg group and the Dedekind zeta function?", "background": "The Birch–Tate conjecture connects algebraic K-theory to number theory. For a number field $F$, it relates the order of the center of the Steinberg group $\\text{St}(\\mathcal{O}_F)$ (where $\\mathcal{O}_F$ is the ring of integers) to special values of the Dedekind zeta function $\\zeta_F(s)$ at $s = -1$. The conjecture predicts that $|\\text{center}(\\text{St}(\\mathcal{O}_F))| = |\\zeta_F(-1)|$ after appropriate normalization. This would provide a deep connection between algebraic structures and analytic number theory, generalizing classical results about class numbers. It fits into the broader Quillen-Lichtenbaum conjecture framework and has implications for understanding higher K-groups.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 245, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1300, "problem_number": "ALG-004", "title": "Crouzeix's Conjecture", "statement": "Is $\\|f(A)\\| \\leq 2\\sup_{z \\in W(A)} |f(z)|$ for all matrices $A$ and functions $f$ analytic on the numerical range?", "background": "Michel Crouzeix conjectured in 2004 that for any $n \\times n$ complex matrix $A$ and any function $f$ analytic on the numerical range $W(A) = \\{\\langle Ax, x \\rangle : \\|x\\| = 1\\}$, the matrix norm satisfies $\\|f(A)\\| \\leq 2\\|f\\|_{W(A)}$. The constant 2 is conjectured to be optimal. Crouzeix proved the bound with constant $11.08$, later improved to $1 + \\sqrt{2} \\approx 2.41$ by various authors. The conjecture is verified for $2 \\times 2$ matrices and special classes. It has applications to functional calculus, matrix functions, and numerical analysis. The problem combines complex analysis, operator theory, and linear algebra, and its resolution would clarify fundamental properties of matrix functions.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 278, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1302, "problem_number": "ALG-006", "title": "Perfect Cuboid", "statement": "Does there exist a rectangular cuboid with integer edges, face diagonals, and space diagonal?", "background": "A perfect cuboid would have integer values for all of: three edge lengths $a, b, c$, three face diagonals $\\sqrt{a^2+b^2}, \\sqrt{b^2+c^2}, \\sqrt{c^2+a^2}$, and the space diagonal $\\sqrt{a^2+b^2+c^2}$. This is the 3D generalization of the Pythagorean triple problem (which has infinitely many solutions). Despite extensive computer searches, no perfect cuboid has been found, nor has impossibility been proven. The problem connects to Diophantine equations, elliptic curves, and number theory. Weaker versions exist: edge-perfect cuboids (all edges and face diagonals integer) are known, as are face-perfect and space-perfect variants. The perfect cuboid is problem D18 in Richard Guy's \"Unsolved Problems in Number Theory\" and has attracted amateur and professional attention for over a century.", "difficulty_level_id": 3, "status": "open", "category_id": 4, "view_count": 423, "favorite_count": 35, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1305, "problem_number": "ALG-009", "title": "Zauner's Conjecture (SIC-POVM)", "statement": "Do symmetric informationally complete POVMs exist in all dimensions?", "background": "Zauner's conjecture, central to quantum information theory, asks whether SIC-POVMs (Symmetric Informationally Complete Positive Operator-Valued Measures) exist in all finite-dimensional Hilbert spaces. A SIC-POVM in dimension $d$ consists of $d^2$ pure quantum states with pairwise fidelity $1/(d+1)$, forming a regular simplex in quantum state space. These structures optimize quantum measurements and have applications in quantum tomography, cryptography, and foundations. SIC-POVMs are known for dimensions up to 193 and many higher dimensions through numerical construction. Analytic constructions exist for infinitely many dimensions using Weyl-Heisenberg groups and number-theoretic methods. The conjecture connects to algebraic number theory (Stark units, ray class fields), representation theory, and Galois theory. Resolving it would clarify fundamental symmetries in quantum mechanics.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 298, "favorite_count": 26, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1308, "problem_number": "ALG-012", "title": "Andrews–Curtis Conjecture", "statement": "Can every balanced presentation of the trivial group be transformed to a trivial presentation by Nielsen moves?", "background": "The Andrews–Curtis conjecture, proposed in 1965, concerns group presentations. A balanced presentation has the same number of generators and relators. The trivial presentation is $\\langle x \\mid x \\rangle$. Nielsen transformations on relators include: replacing relator $r$ with $r^{-1}$, with $rs$ for another relator $s$, or conjugating $r$. The question: can any balanced presentation of the trivial group be reduced to the trivial presentation using these moves? Known counter-examples exist for unbalanced presentations (Rapaport). The conjecture is verified for many cases but remains open in general. It connects to the Zeeman conjecture in topology, 4-manifold theory, and algebraic K-theory. A counter-example would have major implications for understanding fundamental groups and 2-complexes.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 289, "favorite_count": 23, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1310, "problem_number": "ALG-014", "title": "Herzog–Schönheim Conjecture", "statement": "Can a finite system of left cosets forming a partition of a group have distinct indices?", "background": "The Herzog–Schönheim conjecture states: if left cosets $g_iH_i$ of subgroups $H_i$ partition a group $G$, then at least two indices $[G:H_i]$ must be equal. Equivalently, you cannot partition a group using cosets of subgroups with all different indices. The conjecture is verified for many cases: finite abelian groups, free groups, and groups with certain structural properties. It has connections to coverings of groups, number theory (covering congruences—Mycielski's conjecture), and additive combinatorics. The problem appears simple but has resisted general proof. A counter-example would be a group with a highly unusual coset structure, and would impact understanding of group factorizations and tiling problems.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 198, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1318, "problem_number": "ANA-006", "title": "Navier-Stokes Regularity", "statement": "Do smooth initial data for 3D Navier-Stokes equations yield smooth solutions for all time?", "background": "One of the seven Millennium Prize Problems ($1M prize). The 3D Navier-Stokes equations govern fluid flow: $\\partial_t u + (u \\cdot \\nabla)u = \\nu \\Delta u - \\nabla p + f$ with $\\nabla \\cdot u = 0$. Given smooth initial conditions and forcing, do solutions remain smooth globally, or can finite-time singularities develop? In 2D, global regularity is proven. In 3D, existence of weak solutions is known (Leray), but smoothness is open. Partial results establish regularity under smallness conditions or for special data. The problem is central to mathematical fluid dynamics and has deep implications for turbulence, computational fluid dynamics, and the physical validity of the equations. Techniques involve harmonic analysis, functional analysis, and PDE theory.", "difficulty_level_id": 5, "status": "open", "category_id": 9, "view_count": 892, "favorite_count": 67, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1319, "problem_number": "COMB-001", "title": "1/3–2/3 Conjecture", "statement": "Does every non-totally-ordered finite poset have two elements with probability between 1/3 and 2/3 in random linear extensions?", "background": "For a finite partially ordered set (poset) that is not totally ordered, the 1/3–2/3 conjecture asks: do there always exist elements $x$ and $y$ such that the probability $x$ appears before $y$ in a uniformly random linear extension is strictly between 1/3 and 2/3? Linear extensions are total orderings consistent with the partial order. The conjecture was posed in the 1960s and remains open. It has connections to sorting algorithms, computational complexity, and order theory. Known results: true for many special classes of posets, including series-parallel posets. The conjecture would provide insight into the structure of linear extensions and has applications to average-case analysis of sorting and ranking algorithms.", "difficulty_level_id": 4, "status": "open", "category_id": 2, "view_count": 234, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1320, "problem_number": "COMB-002", "title": "Lonely Runner Conjecture", "statement": "If $k$ runners with distinct speeds run on a circular track, will each be lonely (distance $\\geq 1/k$ from others) at some time?", "background": "Proposed by J. M. Wills in 1967, this conjecture concerns runners on a unit-length circular track with distinct constant speeds. A runner is \"lonely\" if all other runners are at distance at least $1/k$ away. The conjecture states every runner is lonely at some time. Verified for $k \\leq 7$ runners. The problem has reformulations in terms of Diophantine approximation, view-obstruction (can $k-1$ points block all views from a point to another on a circle?), and number theory. Applications include scheduling, communication protocols, and chromatic number of certain graphs. Proof techniques use continued fractions, geometry of numbers, and combinatorial arguments. The general case remains stubbornly open despite its elementary statement.", "difficulty_level_id": 4, "status": "open", "category_id": 2, "view_count": 312, "favorite_count": 26, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1321, "problem_number": "COMB-003", "title": "Union-Closed Sets Conjecture", "statement": "For a finite family of sets closed under unions, must some element appear in at least half the sets?", "background": "Frankl's union-closed sets conjecture (1979) states: if a finite family $\\mathcal{F}$ of sets is closed under pairwise unions (i.e., $A, B \\in \\mathcal{F} \\Rightarrow A \\cup B \\in \\mathcal{F}$), then there exists an element appearing in at least $|\\mathcal{F}|/2$ sets. The conjecture is verified for many special cases: families with at most 50 sets, families where the largest set has at most 11 elements, and various structural conditions. In 2024, significant progress was made proving the conjecture holds when relaxing \"half\" to 0.01% (a weakened version). The problem connects to lattice theory, combinatorics, and has reformulations in terms of posets and Boolean functions. A proof would illuminate the structure of union-closed families.", "difficulty_level_id": 4, "status": "open", "category_id": 2, "view_count": 387, "favorite_count": 31, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1322, "problem_number": "COMB-004", "title": "No-Three-in-Line Problem", "statement": "What is the maximum number of points in an $n \\times n$ grid with no three collinear?", "background": "The no-three-in-line problem asks for $g(n)$, the maximum number of points that can be placed in an $n \\times n$ grid such that no three are collinear. Dudeney (1917) conjectured $g(n) = 2n$ for all $n$. Known values: $g(3) = 4, g(4) = 8, g(5) = 10, g(6) = 12$, and computational results extend further. For large $n$, Erdős proved $g(n) \\leq cn/(\\log \\log n)^{1/2}$ for some constant $c$. Lower bounds around $1.85n$ are known. The problem connects to combinatorial geometry, Ramsey theory, and coding theory. Despite its elementary formulation, determining exact values or the asymptotic behavior of $g(n)$ remains challenging. Applications include error-correcting codes and geometric configurations.", "difficulty_level_id": 3, "status": "open", "category_id": 2, "view_count": 298, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1324, "problem_number": "COMB-006", "title": "Sunflower Conjecture", "statement": "For fixed $r$, can the number of size-$k$ sets needed for an $r$-sunflower be bounded by $c^k$ for some constant $c$?", "background": "Erdős and Rado (1960) defined an $r$-sunflower as a collection of $r$ sets $A_1, \\ldots, A_r$ with common intersection $C$ (the core) such that the sets $A_i \\setminus C$ are pairwise disjoint (the petals). Their theorem: any family of size-$k$ sets with at least $k! \\cdot r^k$ members contains an $r$-sunflower. The sunflower conjecture asks: can the bound be improved to $c^k$ for some constant $c = c(r)$ depending only on $r$? This would be sharp up to the value of $c$. In 2019, Alweiss, Lovett, Wu, and Zhang proved a bound of $(\\log k)^k$, a breakthrough improving Erdős-Rado. The conjecture has applications to circuit complexity, learning theory, and DNF formulas. A proof would impact computational complexity theory.", "difficulty_level_id": 4, "status": "open", "category_id": 2, "view_count": 367, "favorite_count": 29, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1327, "problem_number": "GRAPH-003", "title": "Cycle Double Cover Conjecture", "statement": "Does every bridgeless graph have a collection of cycles covering each edge exactly twice?", "background": "The cycle double cover conjecture states: every bridgeless graph (no bridge edges) has a cycle double cover—a collection of cycles such that each edge appears in exactly two cycles. Proposed by Szekeres (1973) and Seymour (1979), this is equivalent to several other conjectures in graph theory. Known for planar graphs (via face boundaries), 4-edge-connected graphs, and graphs with maximum degree at most 3. The conjecture connects to nowhere-zero flows, graph embeddings, and topological graph theory. It would imply results about circular chromatic number and graph decompositions. Despite extensive research, the general case remains open and is considered one of the major problems in graph theory.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 312, "favorite_count": 25, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1328, "problem_number": "GRAPH-004", "title": "Erdős–Hajnal Conjecture", "statement": "For any fixed graph $H$, do $H$-free graphs contain large cliques or independent sets?", "background": "The Erdős–Hajnal conjecture (1977) asks: for any graph $H$, is there $\\delta > 0$ such that every $n$-vertex graph with no induced copy of $H$ contains a clique or independent set of size at least $n^\\delta$? This would dramatically strengthen Ramsey theory for hereditary graph classes. For general graphs, Ramsey theorem gives only polylogarithmic guarantees. The conjecture is proven for specific $H$: paths, trees of bounded diameter, and certain small graphs. Partial results by Alon, Pach, and Solymosi establish weaker bounds. The problem connects to extremal graph theory, Ramsey theory, and structural graph theory. A proof would reveal deep structure in induced-subgraph-free graphs.", "difficulty_level_id": 5, "status": "open", "category_id": 3, "view_count": 289, "favorite_count": 23, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1329, "problem_number": "GRAPH-005", "title": "Lovász Conjecture", "statement": "Does every finite connected vertex-transitive graph have a Hamiltonian path?", "background": "Proposed by László Lovász in 1969, this conjecture states that every finite connected vertex-transitive graph (graph with transitive automorphism group) contains a Hamiltonian path. A stronger version asks for a Hamiltonian cycle. The conjecture is verified for Cayley graphs (Rapaport-Strasser, 1985 for primes; Marušič for certain cases), vertex-transitive graphs of order $pq$ for primes $p < q$, and various special classes. Counter-examples exist for infinite graphs. The problem connects to algebraic graph theory, group theory, and the study of symmetric structures. A proof would significantly advance understanding of Hamiltonian properties in highly symmetric graphs and has implications for network design and routing.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 267, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1330, "problem_number": "GRAPH-006", "title": "Hadwiger–Nelson Problem", "statement": "What is the chromatic number of the plane with unit distance graph coloring?", "background": "The Hadwiger–Nelson problem asks: what is the minimum number of colors needed to color the plane such that no two points at distance exactly 1 have the same color? This is equivalent to finding the chromatic number of the unit distance graph in $\\mathbb{R}^2$. It has been known since 1950 that $4 \\leq \\chi \\leq 7$. In 2018, Aubrey de Grey found a unit distance graph with chromatic number 5, improving the lower bound to 5. The upper bound of 7 uses a hexagonal tiling argument. The exact value is unknown. The problem connects to Euclidean Ramsey theory, discrete geometry, and combinatorial optimization. Extensions to higher dimensions and different metrics are also studied.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 421, "favorite_count": 35, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1331, "problem_number": "TOP-001", "title": "Unknotting Problem", "statement": "Can unknots be recognized in polynomial time?", "background": "The unknotting problem asks whether there exists a polynomial-time algorithm to determine if a given knot diagram represents the unknot (a circle with no actual knots). A knot diagram is a 2D projection of a 3D knot with crossing information. The problem is known to be in NP (a certificate is a sequence of Reidemeister moves) and co-NP (certification via knot invariants). In 2011, Lackenby, building on work by Dynnikov, showed an algorithm exists with complexity bounded by $2^{cn}$ for some constant $c$, where $n$ is the crossing number. However, whether a polynomial-time algorithm exists remains unknown. The problem connects to computational topology, 3-manifold theory, and has applications to molecular biology (DNA unknotting) and physics.", "difficulty_level_id": 4, "status": "open", "category_id": 7, "view_count": 334, "favorite_count": 27, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1332, "problem_number": "TOP-002", "title": "Borel Conjecture", "statement": "Are aspherical closed manifolds determined up to homeomorphism by their fundamental groups?", "background": "The Borel conjecture states: if two aspherical closed manifolds (manifolds with contractible universal cover) have isomorphic fundamental groups, then they are homeomorphic. An aspherical manifold has all higher homotopy groups trivial, so its topology is determined by $\\pi_1$. The conjecture is a topological rigidity statement: algebraic data ($\\pi_1$) determines geometric structure (homeomorphism type). Proven for many special cases: flat manifolds, hyperbolic manifolds (by Mostow rigidity for dimension $\\geq 3$), and certain graph manifolds. The Novikov conjecture is a weaker form (about homotopy invariance of higher signatures). The Borel conjecture connects to surgery theory, K-theory, and geometric group theory.", "difficulty_level_id": 5, "status": "open", "category_id": 7, "view_count": 278, "favorite_count": 22, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1333, "problem_number": "TOP-003", "title": "Volume Conjecture", "statement": "Do quantum invariants of knots relate asymptotically to hyperbolic volume?", "background": "The volume conjecture, proposed by Kashaev (1997) and generalized by Murakami-Murakami, relates quantum topology to hyperbolic geometry. For a hyperbolic knot $K$ in $S^3$, let $J_N(K; q)$ be the colored Jones polynomial at $q = e^{2\\pi i/N}$. The conjecture states: $\\lim_{N \\to \\infty} \\frac{2\\pi \\log|J_N(K; e^{2\\pi i/N})|}{N} = \\text{Vol}(S^3 \\setminus K)$, where the right side is the hyperbolic volume of the knot complement. Verified for many specific knots and families (torus knots, figure-eight). The conjecture suggests deep connections between quantum field theory, Chern-Simons theory, and 3-manifold geometry. It would unify quantum invariants and geometric invariants.", "difficulty_level_id": 5, "status": "open", "category_id": 7, "view_count": 245, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1334, "problem_number": "TOP-004", "title": "Novikov Conjecture", "statement": "Are certain combinations of Pontryagin classes homotopy invariant?", "background": "The Novikov conjecture, proposed by Sergei Novikov in 1965, is a fundamental problem in topology and differential geometry. For a closed oriented manifold $M$ with fundamental group $\\pi$, certain rational linear combinations of Pontryagin classes evaluated on the fundamental class should be homotopy invariants when pushed forward to the classifying space $B\\pi$. More precisely, higher signatures defined using the signature operator should be homotopy invariants. The conjecture is verified for many groups: finite groups, amenable groups, linear groups, Gromov hyperbolic groups, and many others. It connects to K-theory, C*-algebras, index theory, and surgery theory. The conjecture has deep implications for the topology of manifolds and the structure of group C*-algebras.", "difficulty_level_id": 5, "status": "open", "category_id": 7, "view_count": 312, "favorite_count": 25, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1335, "problem_number": "GEOM-007", "title": "Kakeya Conjecture", "statement": "Must a Kakeya set in $\\mathbb{R}^n$ have Hausdorff and Minkowski dimension $n$?", "background": "A Kakeya set in $\\mathbb{R}^n$ is a compact set containing a unit line segment in every direction. The Kakeya conjecture states such sets must have full Hausdorff and Minkowski dimension $n$. In $\\mathbb{R}^2$, Kakeya sets can have measure zero (Davies 1971) but must have Hausdorff dimension 2 (proven). For $n \\geq 3$, the conjecture is open. Known results: Kakeya sets in $\\mathbb{R}^n$ have Hausdorff dimension $\\geq (n+2)/2$ (Wolff, 1995), improved to $\\geq n/2 + \\epsilon$ by various authors. The problem connects to harmonic analysis (Bochner-Riesz conjecture, restriction conjecture), PDE (wave equation estimates), and number theory. Resolving it would impact multiple areas of analysis.", "difficulty_level_id": 5, "status": "open", "category_id": 6, "view_count": 289, "favorite_count": 23, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1336, "problem_number": "GEOM-008", "title": "Illumination Problem", "statement": "Can every convex body in $\\mathbb{R}^n$ be illuminated by $2^n$ light sources?", "background": "The illumination problem (or Hadwiger's problem) asks: what is the minimum number of light sources (point sources or directions) needed to illuminate the entire boundary of any convex body in $\\mathbb{R}^n$? A point on the boundary is illuminated if the ray from the light source to that point does not intersect the interior. Conjecture: $2^n$ sources suffice for dimension $n$. Known results: the upper bound is $\\lfloor 3^{n}/2^{n-1} \\rfloor$ (Schramm), and the conjecture is verified for $n \\leq 3$. The problem connects to discrete geometry, combinatorial geometry, and has applications in computer graphics and sensor placement. A proof would clarify fundamental properties of convex bodies.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 234, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1338, "problem_number": "DYN-002", "title": "MLC Conjecture", "statement": "Is the Mandelbrot set locally connected?", "background": "The MLC (Mandelbrot set is Locally Connected) conjecture asks whether the famous Mandelbrot set—the set of complex parameters $c$ for which the iteration $z_{n+1} = z_n^2 + c$ (starting from $z_0 = 0$) remains bounded—is locally connected. Local connectivity would mean every point has arbitrarily small connected neighborhoods. The conjecture is one of the most important problems in complex dynamics. If true, it would imply: the boundary of the Mandelbrot set has Hausdorff dimension 2, the Mandelbrot set is the closure of its interior, and precise descriptions of the topology. The conjecture has been verified for many parameter regions but remains open in general. It connects to renormalization theory, polynomial dynamics, and fractal geometry.", "difficulty_level_id": 5, "status": "open", "category_id": 9, "view_count": 398, "favorite_count": 32, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1339, "problem_number": "DYN-003", "title": "Weinstein Conjecture", "statement": "Does every regular compact contact-type level set carry a periodic orbit?", "background": "The Weinstein conjecture, proposed by Alan Weinstein in 1978, states: every regular compact contact-type level set of a Hamiltonian on a symplectic manifold carries at least one periodic orbit of the Hamiltonian flow. In more geometric terms, on a compact contact manifold, the Reeb vector field has at least one closed orbit. The conjecture has been proven in many cases: dimension 3 (Taubes, 2007), overtwisted contact 3-manifolds (Hofer, 1993), and various higher-dimensional cases using symplectic field theory and pseudoholomorphic curves. The full conjecture in all dimensions remains open. It connects contact geometry, symplectic topology, Hamiltonian dynamics, and has applications to celestial mechanics and rigid body dynamics.", "difficulty_level_id": 5, "status": "open", "category_id": 9, "view_count": 256, "favorite_count": 20, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1340, "problem_number": "DYN-004", "title": "Birkhoff Conjecture", "statement": "If a billiard table is strictly convex and integrable, must its boundary be an ellipse?", "background": "The Birkhoff conjecture concerns dynamical billiards: if a strictly convex billiard table in the plane is integrable (has a complete set of integrals of motion), then its boundary must be an ellipse. Elliptical billiards are known to be integrable (Birkhoff, 1927). The conjecture asks if they are the only such tables. Partial results: true for sufficiently smooth perturbations of circles and ellipses, and for certain classes of curves. The problem connects to KAM theory, integrable systems, and spectral geometry. A proof would characterize all integrable planar billiards and has implications for understanding caustics, periodic orbits, and the inverse spectral problem.", "difficulty_level_id": 5, "status": "open", "category_id": 9, "view_count": 289, "favorite_count": 23, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1341, "problem_number": "ALGGEOM-001", "title": "Abundance Conjecture", "statement": "If the canonical bundle of a variety is nef, must it be semiample?", "background": "The abundance conjecture is a central problem in birational geometry and minimal model theory. For a projective variety $X$ with Kawamata log terminal singularities, if the canonical bundle $K_X$ is nef (numerically effective—has non-negative intersection with all curves), the conjecture states $K_X$ must be semiample (some positive multiple is globally generated). This would complete the minimal model program by ensuring every minimal model has good positivity properties. Known cases: surfaces (classical), dimension 3 (Miyaoka, Kawamata), toric varieties, and certain special cases in higher dimensions. The conjecture connects to the cone theorem, base point freeness, and would have major implications for classification of algebraic varieties.", "difficulty_level_id": 5, "status": "open", "category_id": 5, "view_count": 234, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1343, "problem_number": "LOGIC-001", "title": "Vaught Conjecture", "statement": "Is the number of countable models of a complete first-order theory finite, $\\aleph_0$, or $2^{\\aleph_0}$?", "background": "The Vaught conjecture, proposed by Robert Vaught in 1961, is a fundamental problem in model theory. For a complete first-order theory in a countable language, the number of countable models (up to isomorphism) must be either finite, countably infinite ($\\aleph_0$), or continuum ($2^{\\aleph_0}$). In other words, there cannot be exactly $\\aleph_1$ (or any other intermediate cardinality) non-isomorphic countable models. The conjecture is known to hold for many classes of theories: $\\omega$-stable theories, superstable theories, and theories with certain structural properties. However, the general case remains open. The problem connects to descriptive set theory, infinitary logic, and has implications for classification theory and the structure of models.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 298, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1344, "problem_number": "LOGIC-002", "title": "Cherlin-Zilber Conjecture", "statement": "Is every simple group with $\\aleph_0$-stable theory an algebraic group over an algebraically closed field?", "background": "The Cherlin-Zilber conjecture concerns the classification of simple groups in model theory. It states: every infinite simple group whose first-order theory is stable in $\\aleph_0$ (countably stable) is isomorphic to a simple algebraic group over an algebraically closed field. The conjecture connects abstract model-theoretic stability to concrete algebraic structures. Many special cases have been verified, and the conjecture has driven development of geometric stability theory. It would provide a complete classification of stable simple groups and has implications for understanding the interaction between model theory and group theory. Zilber's work on Zariski geometries provides evidence for the conjecture.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 245, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1346, "problem_number": "GEOM-009", "title": "Yang-Mills Existence and Mass Gap", "statement": "Does Yang-Mills theory exist mathematically and exhibit a mass gap in 4D?", "background": "One of the seven Millennium Prize Problems ($1M prize). The Yang-Mills equations describe the behavior of elementary particles using non-Abelian gauge theory, fundamental to the Standard Model of particle physics. The problem asks two questions: (1) Does a mathematically rigorous quantum Yang-Mills theory exist in 4-dimensional spacetime? (2) Does it exhibit a mass gap—the smallest mass of any excitation being strictly positive? Physicists use Yang-Mills theory extensively, but a rigorous mathematical foundation is lacking. Proving existence and the mass gap would provide the mathematical basis for quantum chromodynamics (QCD) and explain confinement of quarks. The problem connects quantum field theory, differential geometry, functional analysis, and mathematical physics. Despite extensive physics research, mathematical proof remains elusive.", "difficulty_level_id": 5, "status": "open", "category_id": 6, "view_count": 567, "favorite_count": 47, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1347, "problem_number": "ST-001", "title": "Partition Principle Implies Axiom of Choice", "statement": "Does the partition principle (PP) imply the axiom of choice (AC)?", "background": "The partition principle states that for every partition of a set, there exists a set that contains exactly one element from each cell of the partition. The axiom of choice states that for every collection of nonempty sets, there exists a choice function selecting one element from each set. While AC clearly implies PP, the reverse implication is unknown. This question explores the relative strength of these fundamental axioms in set theory and their role in mathematics.", "difficulty_level_id": 4, "status": "open", "category_id": 10, "view_count": 234, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1348, "problem_number": "ST-002", "title": "Woodin's GCH below Strongly Compact Cardinals", "statement": "Does the generalized continuum hypothesis below a strongly compact cardinal imply it everywhere?", "background": "Posed by W. Hugh Woodin, this problem asks whether local instances of the generalized continuum hypothesis (GCH) can force global instances. A strongly compact cardinal is a large cardinal with strong reflection properties. The question explores whether GCH holding below such a cardinal must propagate throughout the universe of sets. This connects large cardinal theory with cardinal arithmetic and the structure of the set-theoretic universe.", "difficulty_level_id": 5, "status": "open", "category_id": 10, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1349, "problem_number": "ST-003", "title": "GCH and Diamond Principle", "statement": "Does the generalized continuum hypothesis entail the diamond principle $\\diamondsuit(E_{\\text{cf}(\\lambda)}^{\\lambda^+})$ for every singular cardinal $\\lambda$?", "background": "The diamond principle is a combinatorial principle asserting the existence of certain prediction sequences. For singular cardinals (cardinals not equal to their own cofinality), the relationship between GCH and diamond principles is subtle. While diamond holds at successor cardinals under GCH, its behavior at successors of singular cardinals remains mysterious. This problem probes the fine structure of cardinal arithmetic.", "difficulty_level_id": 5, "status": "open", "category_id": 10, "view_count": 156, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1350, "problem_number": "ST-004", "title": "GCH and Suslin Trees", "statement": "Does the generalized continuum hypothesis imply the existence of an $\\aleph_2$-Suslin tree?", "background": "A Suslin tree is a tree of height $\\omega_1$ with no uncountable chains or antichains. An $\\aleph_2$-Suslin tree is the analogous structure at the next cardinal level. While Suslin trees at $\\aleph_1$ can exist under certain axioms, their existence at $\\aleph_2$ under GCH is unknown. This problem connects cardinal arithmetic with combinatorial set theory and the theory of infinite trees.", "difficulty_level_id": 4, "status": "open", "category_id": 10, "view_count": 167, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1352, "problem_number": "ST-006", "title": "Ultimate Core Model", "statement": "Does there exist an ultimate core model containing all large cardinals?", "background": "Core models are canonical inner models of set theory that approximate the entire universe while being more tractable. The search for an ultimate core model—one encompassing all large cardinal properties—is a central goal of modern set theory. Such a model would unify our understanding of large cardinals and provide a framework for resolving independence questions. The project involves deep interactions between forcing, inner model theory, and large cardinal axioms.", "difficulty_level_id": 5, "status": "open", "category_id": 10, "view_count": 178, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1353, "problem_number": "ST-007", "title": "Woodin's Ω-Conjecture", "statement": "If there is a proper class of Woodin cardinals, does Ω-logic satisfy an analogue of Gödel's completeness theorem?", "background": "Proposed by W. Hugh Woodin, this conjecture connects large cardinals with logic. Ω-logic is a strong logic using Woodin cardinals to define semantic validity. The conjecture asserts that under the assumption of a proper class of Woodin cardinals, Ω-logic becomes complete in a generalized sense—every Ω-valid sentence has an Ω-proof. This would provide a powerful new framework for set-theoretic truth and resolve many independence questions.", "difficulty_level_id": 5, "status": "open", "category_id": 10, "view_count": 145, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1354, "problem_number": "ST-008", "title": "Strongly Compact vs Supercompact Cardinals", "statement": "Does the consistency of a strongly compact cardinal imply the consistent existence of a supercompact cardinal?", "background": "Strongly compact cardinals and supercompact cardinals are both large cardinal notions with powerful reflection properties. Supercompact cardinals are known to be stronger, but whether their consistency strength is strictly greater than strongly compact cardinals remains open. This problem probes the fine structure of the large cardinal hierarchy and the relationships between different reflection principles.", "difficulty_level_id": 5, "status": "open", "category_id": 10, "view_count": 167, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1355, "problem_number": "ST-009", "title": "Jónsson Algebra on ℵ_ω", "statement": "Does there exist a Jónsson algebra on $\\aleph_\\omega$?", "background": "A Jónsson algebra is an algebraic structure with no proper subalgebra of the same cardinality. The existence of Jónsson algebras on various cardinals connects algebra with set theory. For $\\aleph_\\omega$ (the $\\omega$-th infinite cardinal), existence remains unknown. A positive answer would provide new insights into the algebraic structure of infinite sets and the behavior of singular cardinals.", "difficulty_level_id": 4, "status": "open", "category_id": 10, "view_count": 134, "favorite_count": 10, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1356, "problem_number": "ST-010", "title": "Open Coloring Axiom and Continuum Hypothesis", "statement": "Is the open coloring axiom (OCA) consistent with $2^{\\aleph_0} > \\aleph_2$?", "background": "The open coloring axiom is a combinatorial principle with powerful consequences for the structure of the real line. It is known to be consistent with $2^{\\aleph_0} = \\aleph_2$, but consistency with larger values of the continuum is unknown. This problem explores the interaction between partition properties and cardinal arithmetic, central themes in modern set theory.", "difficulty_level_id": 4, "status": "open", "category_id": 10, "view_count": 156, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1357, "problem_number": "ST-011", "title": "Reinhardt Cardinals without Choice", "statement": "Without assuming the axiom of choice, can a nontrivial elementary embedding V→V exist?", "background": "A Reinhardt cardinal would witness an elementary embedding from the universe of all sets (V) to itself. Kunen proved such embeddings cannot exist with the axiom of choice. However, without AC, the question remains open. Reinhardt cardinals would be the strongest large cardinal notion, transcending the usual hierarchy. Their possible existence connects to alternative set theories and the role of choice in mathematics.", "difficulty_level_id": 5, "status": "open", "category_id": 10, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1358, "problem_number": "GAME-001", "title": "Sudoku: Unique Solution Puzzles", "statement": "How many Sudoku puzzles have exactly one solution?", "background": "Standard 9×9 Sudoku grids can be filled in approximately 6.67 × 10²¹ ways. A puzzle is a partial filling with a unique completion. Despite extensive computer searches, the exact count of puzzles with unique solutions remains unknown. This combinatorial problem involves constraints, symmetry breaking, and counting techniques. Understanding this would illuminate the mathematical structure underlying Sudoku and related constraint satisfaction problems.", "difficulty_level_id": 2, "status": "open", "category_id": 2, "view_count": 892, "favorite_count": 67, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1359, "problem_number": "GAME-002", "title": "Sudoku: Minimal Puzzles Count", "statement": "How many Sudoku puzzles with exactly one solution are minimal (removing any clue creates multiple solutions)?", "background": "A minimal Sudoku puzzle cannot have any clue removed without losing uniqueness. While we know examples with as few as 17 clues, the total count of minimal puzzles is unknown. This problem combines enumeration with the structure of constraint systems. The answer would deepen our understanding of puzzle difficulty, minimal representations, and the geometry of solution spaces.", "difficulty_level_id": 2, "status": "open", "category_id": 2, "view_count": 678, "favorite_count": 51, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1360, "problem_number": "GAME-003", "title": "Maximum Givens in Minimal Sudoku", "statement": "What is the maximum number of givens for a minimal Sudoku puzzle?", "background": "While minimal puzzles can have as few as 17 givens, the upper bound is unknown. A puzzle with many givens can still be minimal if each clue is essential. Computer searches have found minimal puzzles with around 40 givens, but no theoretical maximum is known. This question explores the relationship between redundancy, minimality, and constraint propagation in combinatorial problems.", "difficulty_level_id": 2, "status": "open", "category_id": 2, "view_count": 567, "favorite_count": 43, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1361, "problem_number": "GAME-004", "title": "Tic-Tac-Toe Winning Dimension", "statement": "Given the width of a tic-tac-toe board, what is the smallest dimension guaranteeing X has a winning strategy?", "background": "Classic tic-tac-toe is a draw with perfect play. In higher dimensions (n^d game), questions become more complex. The Hales-Jewett theorem guarantees that for any fixed line length n, there exists a dimension d where the first player wins. But finding the exact threshold dimension for each n remains open. This connects combinatorics, game theory, and Ramsey theory.", "difficulty_level_id": 3, "status": "open", "category_id": 2, "view_count": 445, "favorite_count": 34, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1362, "problem_number": "GAME-005", "title": "Perfect Chess", "statement": "What is the outcome of a perfectly played game of chess?", "background": "Chess is a finite deterministic game, so theoretically one of three outcomes holds with perfect play: White wins, Black wins, or draw. Despite centuries of play and powerful computers, we don't know which. The game tree has approximately 10⁴⁷ positions, far beyond exhaustive analysis. Current evidence suggests a draw, but proving it requires breakthrough techniques in game-tree search, endgame databases, or mathematical analysis of chess positions.", "difficulty_level_id": 3, "status": "open", "category_id": 2, "view_count": 1534, "favorite_count": 112, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1363, "problem_number": "GAME-006", "title": "Perfect Komi in Go", "statement": "What is the perfect value of komi (compensation points) in Go?", "background": "In Go, komi compensates the second player (White) for Black's first-move advantage. Professional play uses 6.5 or 7.5 points. But what value makes the game perfectly fair with optimal play? Go's complexity (10¹⁷⁰ legal positions) prevents exhaustive analysis. AI like AlphaGo suggest small adjustments, but perfect komi remains unknown. Determining it would require solving Go—understanding the game-theoretic value with perfect play.", "difficulty_level_id": 3, "status": "open", "category_id": 2, "view_count": 789, "favorite_count": 58, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1364, "problem_number": "GAME-007", "title": "Cap Set Problem", "statement": "What is the largest possible cap set in $n$-dimensional affine space over the three-element field?", "background": "A cap set is a collection of points with no three in a line (in the game SET, cards with no valid set). In the affine space $\\mathbb{F}_3^n$, the maximum cap set size is conjectured to be $c^n$ for some constant c < 3. The best bounds are $2.756^n$ (Ellenberg-Gijswijt, 2016). Determining the precise growth rate connects additive combinatorics, polynomial methods, and the cap set conjecture. The breakthrough proof technique revolutionized the field.", "difficulty_level_id": 4, "status": "open", "category_id": 2, "view_count": 356, "favorite_count": 28, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1365, "problem_number": "GAME-008", "title": "Octal Games Periodicity", "statement": "Are the nim-sequences of all finite octal games eventually periodic?", "background": "Octal games are impartial combinatorial games defined by simple rules encoded in octal notation. Their nim-values (Grundy numbers) determine optimal play. For some octal games, the nim-sequence is eventually periodic; for others, patterns are elusive. Whether all finite octal games have eventually periodic nim-sequences is unknown. This problem connects game theory, number theory, and automata theory.", "difficulty_level_id": 3, "status": "open", "category_id": 2, "view_count": 234, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1366, "problem_number": "GAME-009", "title": "Grundy's Game Periodicity", "statement": "Is the nim-sequence of Grundy's game eventually periodic?", "background": "Grundy's game: split a heap of n beans into two unequal heaps; last player to move wins. The nim-value sequence starts 0,1,0,2,1,3,2,1,0,4,... but no period has been found despite extensive computation. Whether it's eventually periodic (or even computable) is open. This specific game has resisted analysis for decades, representing a frontier in combinatorial game theory.", "difficulty_level_id": 3, "status": "open", "category_id": 2, "view_count": 278, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1367, "problem_number": "GAME-010", "title": "Rendezvous Problem", "statement": "What is the optimal strategy for two agents to meet on a network without communication?", "background": "The rendezvous problem asks: how should two agents move on a graph to minimize expected meeting time, when they can't communicate and may not know the graph structure? Variants include symmetric/asymmetric information, labeled/unlabeled nodes, and different graph families. Optimal strategies are known for some simple cases but remain open for general graphs. This problem bridges game theory, probability, and distributed algorithms.", "difficulty_level_id": 3, "status": "open", "category_id": 2, "view_count": 312, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1369, "problem_number": "GEOM-010", "title": "Kissing Number Problem", "statement": "What is the kissing number (maximum number of non-overlapping unit spheres that can touch a central unit sphere) in dimensions other than 1, 2, 3, 4, 8, and 24?", "background": "The kissing number is known exactly only in dimensions 1 (2), 2 (6), 3 (12), 4 (24), 8 (240), and 24 (196,560). The problem asks for exact values in other dimensions. In dimension 3, twelve spheres can kiss a central sphere (with centers forming an icosahedron). Dimensions 8 and 24 have exceptional symmetries related to E₈ and the Leech lattice. Determining kissing numbers connects sphere packing, coding theory, and discrete geometry. The problem is surprisingly difficult—even dimension 5 remains unsolved.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 534, "favorite_count": 41, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1372, "problem_number": "GEOM-013", "title": "Tammes Problem", "statement": "For n > 14 points (except n=24), what is the maximum minimum distance between points on a unit sphere?", "background": "The Tammes problem asks: how should n points be arranged on a sphere to maximize the minimum distance between any pair? This is equivalent to packing n spherical caps on a sphere. Solutions are known for n ≤ 14 and n = 24 (related to exceptional geometries). For other n, only bounds and computational results exist. The problem has applications in molecular chemistry (electron repulsion), coding theory, and crystallography. Named after Dutch botanist who studied pollen grain pores.", "difficulty_level_id": 3, "status": "open", "category_id": 6, "view_count": 245, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1373, "problem_number": "GEOM-014", "title": "Carathéodory Conjecture", "statement": "Does every convex, closed, twice-differentiable surface in 3D Euclidean space have at least two umbilical points?", "background": "An umbilical point on a surface is where the two principal curvatures are equal (the surface curves equally in all directions, like on a sphere). Carathéodory conjectured that every smooth convex closed surface must have at least two umbilic points. A sphere has infinitely many (every point), but most surfaces should have at least two. Despite being over 100 years old, the conjecture remains open. Partial results exist for analytic surfaces and surfaces with special symmetries.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 312, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1374, "problem_number": "GEOM-015", "title": "Cartan-Hadamard Conjecture", "statement": "Does the isoperimetric inequality extend to Cartan-Hadamard manifolds (complete simply-connected manifolds of nonpositive curvature)?", "background": "The classical isoperimetric inequality states that among all regions with fixed perimeter in Euclidean space, the circle (or sphere) encloses maximum area (or volume). The Cartan-Hadamard conjecture asks whether this inequality holds in spaces of nonpositive curvature. Proven in dimensions 2, 3, and 4, but open in higher dimensions. A positive answer would show that negative curvature preserves this fundamental geometric optimization principle.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 267, "favorite_count": 20, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1375, "problem_number": "GEOM-016", "title": "Chern's Conjecture (Affine Geometry)", "statement": "Does the Euler characteristic of a compact affine manifold vanish?", "background": "An affine manifold is a manifold with an atlas whose transition functions are affine transformations. Chern conjectured that any closed (compact, boundaryless) affine manifold must have Euler characteristic zero. The conjecture is true for many special cases but remains open in general. This would be a fundamental constraint on the topology of spaces admitting flat affine structures, connecting differential geometry with algebraic topology.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1376, "problem_number": "GEOM-017", "title": "Hopf Conjectures", "statement": "What are the relationships between curvature and Euler characteristic for higher-dimensional Riemannian manifolds?", "background": "The Hopf conjectures are a collection of problems relating the curvature of a manifold to its Euler characteristic. One version: does a positively curved even-dimensional manifold have positive Euler characteristic? Another: does a negatively curved manifold have zero Euler characteristic? These would generalize the Gauss-Bonnet theorem to higher dimensions. Despite progress on special cases, the general conjectures remain open, representing a frontier in global differential geometry.", "difficulty_level_id": 5, "status": "open", "category_id": 6, "view_count": 234, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1377, "problem_number": "GEOM-018", "title": "Yau's Conjecture on First Eigenvalue", "statement": "Is the first eigenvalue of the Laplace-Beltrami operator on an embedded minimal hypersurface of $S^{n+1}$ equal to $n$?", "background": "This conjecture by Shing-Tung Yau concerns minimal surfaces (soap-film-like surfaces) embedded in spheres. The Laplace-Beltrami operator generalizes the Laplacian to curved spaces. Yau conjectured that the first eigenvalue equals the dimension n for minimal hypersurfaces in the (n+1)-sphere. This would provide a sharp geometric-spectral inequality, connecting the shape of minimal surfaces to their vibration modes. Proven in special cases, but remains open generally.", "difficulty_level_id": 5, "status": "open", "category_id": 6, "view_count": 178, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1378, "problem_number": "GEOM-019", "title": "Hadwiger Conjecture (Covering)", "statement": "Can every $n$-dimensional convex body be covered by at most $2^n$ smaller positively homothetic copies?", "background": "Hadwiger conjectured that any convex body in n dimensions can be covered by at most 2ⁿ smaller copies that are scaled-down versions (homotheties with positive ratio). Proven only for n ≤ 3. For n=2, four copies suffice (proven by Levi). For n=3, eight copies suffice (Hadwiger's original proof). Higher dimensions remain completely open. This is one of the most important unsolved problems in convex geometry, with connections to Borsuk's problem and covering theory.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 298, "favorite_count": 23, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1379, "problem_number": "GEOM-020", "title": "Happy Ending Problem", "statement": "What is the minimum number $g(n)$ of points in general position in the plane guaranteeing a convex $n$-gon?", "background": "The Happy Ending problem (named for the romance between Erdős and Szekeres who solved special cases) asks: how many points in general position (no three collinear) force the existence of n points forming a convex n-gon? Known: g(3)=3, g(4)=5, g(5)=9. Erdős-Szekeres proved $2^{n-2} + 1 \\leq g(n) \\leq \\binom{2n-4}{n-2} + 1$. The exact value for n ≥ 6 is unknown, and closing this exponential gap is a major challenge in combinatorial geometry.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 345, "favorite_count": 27, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1380, "problem_number": "GEOM-021", "title": "Heilbronn Triangle Problem", "statement": "What configuration of $n$ points in the unit square maximizes the area of the smallest triangle they determine?", "background": "Heilbronn asked: place n points in a unit square to maximize the minimum triangle area. Trivially, the minimum area is ≤ 2/n. Heilbronn conjectured it's O(1/n²). Komlos-Pintz-Szemeredi showed it's actually Θ((log n)/n²), disproving the conjecture. However, the exact constant is unknown, and tight bounds remain elusive. This problem exemplifies how discrete geometry problems can have surprising answers and connects to irregularities of distribution and discrepancy theory.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 223, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1381, "problem_number": "GEOM-022", "title": "Kalai's 3^d Conjecture", "statement": "Does every centrally symmetric $d$-dimensional polytope have at least $3^d$ faces?", "background": "Gil Kalai conjectured that centrally symmetric polytopes (symmetric under reflection through the origin) must have many faces—at least 3^d for dimension d. The d-cube achieves this bound exactly. Proved for d ≤ 4. Higher dimensions remain open. This would be a fundamental constraint on the combinatorial complexity of symmetric polytopes, with implications for optimization, linear programming, and the geometry of convex bodies.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1382, "problem_number": "GEOM-023", "title": "Orchard-Planting Problem", "statement": "What is the maximum number of 3-point lines attainable by a configuration of $n$ points in the plane?", "background": "An orchard-planting problem asks: arrange n points (trees) to maximize the number of lines containing exactly 3 points (rows). For n points, at most n(n-1)/6 such lines are possible (by counting). Some configurations achieve this bound or come close. The problem asks for the exact maximum for each n. Solutions are known for small n, but the general pattern is mysterious. This connects to projective geometry, matroid theory, and combinatorial designs.", "difficulty_level_id": 3, "status": "open", "category_id": 6, "view_count": 234, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1383, "problem_number": "GEOM-024", "title": "Unit Distance Problem", "statement": "How many pairs of points at unit distance can be determined by $n$ points in the Euclidean plane?", "background": "Erdős asked: what's the maximum number of unit-distance pairs among n points in the plane? Trivially at most n(n-1)/2. Known: the maximum is Θ(n^(4/3)) (lower bound by Erdős, upper by Spencer-Szemerédi-Trotter). But the exact exponent is unknown—it could be n^(4/3), n^(3/2), or something between. Determining this connects incidence geometry, graph theory, and the crossing number. The unit distance graph has fascinating properties.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 267, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1384, "problem_number": "GEOM-025", "title": "Bellman's Lost-in-a-Forest Problem", "statement": "What is the shortest path that guarantees reaching the boundary of a given shape, starting from an unknown point with unknown orientation?", "background": "You're lost in a forest (a region with known shape but unknown location and orientation). What path guarantees you'll reach the edge? For a circle of radius 1, a path of length ≤ 2 + π/3 ≈ 3.05 suffices. For a square, the answer is unknown. For general convex regions, the problem is wide open. This classic problem in geometric search theory has applications to robotics, navigation, and computational geometry. Finding optimal escape paths connects geometry with optimization.", "difficulty_level_id": 3, "status": "open", "category_id": 6, "view_count": 423, "favorite_count": 33, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1385, "problem_number": "GEOM-026", "title": "Borromean Rings Question", "statement": "Can three unknotted space curves (not all circles) be arranged as Borromean rings?", "background": "Borromean rings are three linked loops where removing any one unlinks the other two. Classical Borromean rings use circles, but perfect circular realization is impossible (proved). Can non-circular unknotted curves realize this linking pattern? This question connects knot theory, topology, and geometry. While Borromean rings can be made from ellipses or other shapes, whether three genuinely unknotted (topologically circular) but geometrically non-circular curves can achieve this remains subtle.", "difficulty_level_id": 3, "status": "open", "category_id": 6, "view_count": 312, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1386, "problem_number": "GEOM-027", "title": "Danzer's Problem", "statement": "Do Danzer sets of bounded density or bounded separation exist?", "background": "A Danzer set is a set of points in the plane such that every convex region of area 1 contains at least one point. Danzer asked: can such a set have bounded density (points per unit area) or bounded separation (minimum distance between points)? Both properties would mean the points are \"well-distributed.\" While Danzer sets exist, whether nice ones exist is open. Related to Conway's \"dead fly\" problem. Connects measure theory, convexity, and geometric covering.", "difficulty_level_id": 4, "status": "open", "category_id": 6, "view_count": 201, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1388, "problem_number": "GRAPH-001", "title": "Brouwer's Conjecture on Graph Laplacians", "statement": "Can the sum of eigenvalues of the Laplacian matrix of a graph be bounded by the number of edges?", "background": "Brouwer conjectured an upper bound for the sum of the k largest eigenvalues of the Laplacian matrix of a graph in terms of the number of edges. The Laplacian matrix encodes graph structure and has deep connections to spectral graph theory. This conjecture would provide fundamental insights into the relationship between a graph's combinatorial and spectral properties. Progress has been made for special classes of graphs, but the general case remains open.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 234, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1389, "problem_number": "GRAPH-002", "title": "Eternal Domination vs Domination Number", "statement": "Does there exist a graph where the dominating number equals the eternal dominating number and both are less than the clique covering number?", "background": "The dominating number γ(G) is the minimum size of a dominating set. The eternal dominating number γ∞(G) arises from a game where guards on vertices must respond to attacks. The question asks if these can equal each other while being smaller than the clique covering number (minimum number of cliques needed to cover all vertices). This connects domination theory with graph games and clique structures.", "difficulty_level_id": 3, "status": "open", "category_id": 3, "view_count": 156, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1390, "problem_number": "GRAPH-003", "title": "Graham's Pebbling Conjecture", "statement": "Is the pebbling number of the Cartesian product of two graphs at least the product of their pebbling numbers?", "background": "Graph pebbling is a combinatorial game where pebbles are moved on vertices according to specific rules. Graham conjectured that the pebbling number (minimum pebbles needed to guarantee placing one on any target vertex) of a Cartesian product G × H is at least π(G) × π(H). Despite progress on special cases like products with paths or cycles, the general conjecture remains unsolved. This problem has applications to communication networks and resource distribution.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1391, "problem_number": "GRAPH-004", "title": "Meyniel's Conjecture on Cop Number", "statement": "Is the cop number of a connected n-vertex graph $O(\\sqrt{n})$?", "background": "The cop number is the minimum number of cops needed to guarantee catching a robber in a pursuit game on a graph. Meyniel conjectured that for any connected graph with n vertices, the cop number is at most O(√n). The best known upper bound is O(n/log n). This problem connects graph theory with algorithmic game theory and has applications to network security, robot motion planning, and pursuit-evasion games. Resolving it would fundamentally advance our understanding of graph searching problems.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 267, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1392, "problem_number": "GRAPH-005", "title": "Graph Coloring Game Monotonicity", "statement": "If Alice has a winning strategy for the vertex coloring game with k colors, does she have one for k+1 colors?", "background": "In the graph coloring game, two players alternately color vertices with k colors, trying to create (Alice) or avoid (Bob) a proper coloring. Intuitively, having more colors should make Alice's task easier. However, whether winning with k colors implies winning with k+1 colors is surprisingly still open. This problem probes the subtle complexity of graph coloring games and connects combinatorial game theory with chromatic graph theory.", "difficulty_level_id": 3, "status": "open", "category_id": 3, "view_count": 178, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1393, "problem_number": "GRAPH-006", "title": "1-Factorization Conjecture", "statement": "Does every k-regular graph on 2n vertices admit a 1-factorization when k ≥ n (or k ≥ n-1 for even n)?", "background": "A 1-factor is a perfect matching, and a 1-factorization is a partition of edges into 1-factors. The conjecture states that sufficiently regular graphs can be decomposed into perfect matchings. This would generalize classical results on complete graphs. Proven for many special cases, but the general statement remains open. Applications include tournament scheduling, network routing, and combinatorial designs.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 201, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1394, "problem_number": "GRAPH-007", "title": "Perfect 1-Factorization Conjecture", "statement": "Does every complete graph on an even number of vertices admit a perfect 1-factorization?", "background": "A perfect 1-factorization of a complete graph K₂ₙ is a 1-factorization where the union of any two 1-factors forms a Hamiltonian cycle. Such structures have beautiful symmetry and applications to combinatorial designs. While perfect 1-factorizations are known for many n (especially powers of 2 and small cases), a general existence proof remains elusive. This is one of the most elegant open problems in graph decomposition theory.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 234, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1395, "problem_number": "GRAPH-008", "title": "Cereceda's Conjecture", "statement": "For k-degenerate graphs, can any (k+2)-coloring be transformed to any other in polynomial steps via single-vertex recolorings?", "background": "Cereceda's conjecture concerns the diameter of the reconfiguration graph of graph colorings. It asks whether the shortest sequence of single-vertex recolorings transforming one coloring to another is polynomially bounded for degenerate graphs. This connects graph coloring with reconfiguration problems—a growing area studying how to transform one solution to another. Applications include network reconfiguration and state-space search.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 167, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1396, "problem_number": "GRAPH-009", "title": "Earth-Moon Problem", "statement": "What is the maximum chromatic number of biplanar graphs?", "background": "A graph is biplanar if it can be drawn on two parallel planes (Earth and Moon) with edges possibly crossing between planes but not within each plane. The Earth-Moon problem asks for the maximum chromatic number of such graphs. Known bounds are 12 ≤ χ ≤ 16. This problem combines planarity concepts with multilayer graph drawings, relevant to VLSI design and network visualization.", "difficulty_level_id": 3, "status": "open", "category_id": 3, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1397, "problem_number": "GRAPH-010", "title": "Gyárfás-Sumner Conjecture", "statement": "Is every graph class defined by excluding one fixed tree as an induced subgraph χ-bounded?", "background": "A graph class is χ-bounded if there's a function f such that every graph in the class with clique number ω has chromatic number at most f(ω). The conjecture states that forbidding any tree as an induced subgraph creates a χ-bounded class. This would unify many results on perfect graphs and their generalizations. The conjecture connects structural graph theory with coloring, and has implications for algorithmic graph theory.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 178, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1398, "problem_number": "GRAPH-011", "title": "Jaeger's Petersen Coloring Conjecture", "statement": "Does every bridgeless cubic graph have a cycle-continuous mapping to the Petersen graph?", "background": "Jaeger conjectured that every bridgeless cubic graph admits a special kind of homomorphism to the Petersen graph that preserves cycle structure. The Petersen graph plays a central role in graph theory as a universal counterexample and fundamental object. This conjecture connects graph homomorphisms, snarks (cubic graphs resistant to edge coloring), and the structure of cubic graphs. It has deep implications for edge coloring and nowhere-zero flow problems.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 156, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1399, "problem_number": "GRAPH-012", "title": "List Coloring Conjecture", "statement": "For every graph, does the list chromatic index equal the chromatic index?", "background": "The chromatic index χ'(G) is the minimum number of colors needed to color edges so no two adjacent edges share a color. The list chromatic index is the minimum k such that edges can be colored from arbitrary k-element color lists. The conjecture states these are always equal. While proven for bipartite graphs and some other classes, the general case remains open. This is a fundamental question in list coloring theory.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 198, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1400, "problem_number": "GRAPH-013", "title": "Overfull Conjecture", "statement": "Is a graph with maximum degree Δ(G) ≥ n/3 in class 2 if and only if it has an overfull subgraph with the same maximum degree?", "background": "By Vizing's theorem, every graph has chromatic index Δ or Δ+1 (class 1 or 2). A graph is overfull if it has more than Δ⌊n/2⌋ edges, forcing class 2. The overfull conjecture provides a complete characterization: when Δ ≥ n/3, being class 2 is equivalent to having an overfull subgraph preserving the maximum degree. This would elegantly explain why graphs are hard to edge-color.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 167, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1401, "problem_number": "GRAPH-014", "title": "Total Coloring Conjecture", "statement": "Is the total chromatic number of every graph at most Δ + 2, where Δ is the maximum degree?", "background": "Total coloring requires coloring both vertices and edges so adjacent/incident elements have different colors. Behzad and Vizing independently conjectured that the total chromatic number χ″(G) ≤ Δ(G) + 2. The lower bound Δ + 1 is easy (color each vertex and its incident edges distinctly). The upper bound Δ + 2 is proven for many graph classes but remains open in general. This is one of the most fundamental open problems in graph coloring.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 245, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1402, "problem_number": "GRAPH-015", "title": "Albertson Conjecture", "statement": "Can the crossing number of a graph be lower-bounded by the crossing number of a complete graph with the same chromatic number?", "background": "The crossing number is the minimum number of edge crossings in a planar drawing. Albertson conjectured cr(G) ≥ cr(K_χ(G)) where χ(G) is the chromatic number. This would link two fundamental graph parameters—crossing number and chromatic number. Proven for chromatic numbers up to 16, but the general case remains open. This connects graph drawing, coloring theory, and topological graph theory.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 178, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1403, "problem_number": "GRAPH-016", "title": "Conway's Thrackle Conjecture", "statement": "Does every thrackle have at most as many edges as vertices?", "background": "A thrackle is a graph drawing where every pair of edges either meets at a common vertex or crosses exactly once. Conway conjectured that thrackles satisfy |E| ≤ |V|. Despite looking simple, this conjecture has resisted proof for decades. The best known bound is |E| ≤ 3|V|/2. This problem connects graph drawing with combinatorial geometry and has surprising depth for such a simply stated question.", "difficulty_level_id": 3, "status": "open", "category_id": 3, "view_count": 201, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1404, "problem_number": "GRAPH-017", "title": "GNRS Conjecture", "statement": "Do minor-closed graph families have $\\ell_1$ embeddings with bounded distortion?", "background": "The GNRS conjecture asks whether graphs from minor-closed families (like planar graphs) can be embedded into L₁ space (ℓ₁ metric) with distortion bounded by a function of the excluded minor size. This connects graph theory with metric geometry and theoretical computer science. The conjecture has important implications for approximation algorithms and understanding the metric structure of graph families.", "difficulty_level_id": 5, "status": "open", "category_id": 3, "view_count": 145, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1405, "problem_number": "GRAPH-018", "title": "Harborth's Conjecture", "statement": "Can every planar graph be drawn with integer edge lengths?", "background": "Harborth conjectured that every planar graph has a straight-line drawing where all edge lengths are integers. While planar graphs always have straight-line drawings (Fáry's theorem), forcing integer lengths is much harder. Known for trees and some other classes, but open in general. This problem connects graph drawing with discrete geometry and has applications to VLSI layout.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1406, "problem_number": "GRAPH-019", "title": "Negami's Conjecture", "statement": "Does every graph with a planar cover have a projective-plane embedding?", "background": "Negami conjectured that if a graph G has a planar cover (a planar graph that maps onto G), then G embeds in the projective plane. This would characterize projective-plane graphs via covering spaces. The conjecture connects topological graph theory with covering space theory from topology. Despite progress on special cases, the general conjecture remains a central open problem in topological graph theory.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 156, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1407, "problem_number": "GRAPH-020", "title": "Turán's Brick Factory Problem", "statement": "What is the minimum crossing number of the complete bipartite graph $K_{m,n}$?", "background": "Turán's brick factory problem asks for the exact crossing number of complete bipartite graphs K_{m,n}. Zarankiewicz conjectured a formula in 1954, which is known to be correct for several cases but unproven in general. The problem arose from Turán observing workers crossing paths while moving bricks. Despite being simple to state, this geometric problem has remained unsolved for 70 years.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 212, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1408, "problem_number": "GRAPH-021", "title": "Guy's Crossing Number Conjecture", "statement": "Is the crossing number of the complete graph $K_n$ equal to the value given by Guy's formula?", "background": "Guy conjectured a formula for the crossing number of complete graphs: cr(K_n) = (1/4)⌊n/2⌋⌊(n-1)/2⌋⌊(n-2)/2⌋⌊(n-3)/2⌋. This is proven for n ≤ 12, but the general case is open. Finding the exact crossing number of complete graphs is a fundamental problem in topological graph theory. The conjecture represents our best guess based on known constructions.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 198, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1409, "problem_number": "GRAPH-022", "title": "Universal Point Sets", "statement": "Do planar graphs have universal point sets of subquadratic size?", "background": "A universal point set for n-vertex planar graphs is a set of points such that every n-vertex planar graph has a straight-line embedding on these points. Trivially, O(n²) points suffice. The question asks if o(n²) is possible. Best known lower bound is Ω(n), upper bound is O(n²). Closing this gap would advance our understanding of planar graph representations and geometric graph theory.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 167, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1410, "problem_number": "GRAPH-023", "title": "Conference Graph Existence", "statement": "Does there exist a conference graph for every number of vertices $v > 1$ where $v \\equiv 1 \\pmod{4}$ and v is an odd sum of two squares?", "background": "A conference graph is a strongly regular graph with specific parameters related to conference matrices. The existence question for these graphs connects graph theory with number theory (sums of squares) and design theory. Known to exist for many values, but a complete characterization remains elusive. These graphs have applications in coding theory and experimental design.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 145, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1411, "problem_number": "GRAPH-024", "title": "Conway's 99-Graph Problem", "statement": "Does there exist a strongly regular graph with parameters (99,14,1,2)?", "background": "Conway asked whether a strongly regular graph with these specific parameters exists. The parameters pass all known necessary conditions (feasibility, integrality), but no construction is known. This is the smallest open case for strongly regular graphs. Finding such a graph or proving nonexistence would advance our understanding of the constraints on strongly regular graphs beyond the known necessary conditions.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 178, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1412, "problem_number": "GRAPH-025", "title": "Degree Diameter Problem", "statement": "For given maximum degree d and diameter k, what is the largest possible number of vertices in a graph?", "background": "The degree diameter problem asks for the maximum order (number of vertices) of a graph with maximum degree d and diameter k. The Moore bound provides an upper limit, but it's rarely achieved (only for very special parameters). Finding the exact values or better bounds is a central problem in extremal graph theory with applications to network design. Tables of best known values are maintained, but many cases remain unsolved.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1413, "problem_number": "GRAPH-026", "title": "Moore Graph Existence", "statement": "Does a Moore graph with girth 5 and degree 57 exist?", "background": "Moore graphs are extremal graphs achieving the Moore bound—the maximum possible vertices for given degree and diameter. The Hoffman-Singleton theorem shows Moore graphs with girth 5 can only have degree 2, 3, 7, or possibly 57. Graphs for degrees 2, 3, 7 are known (cycle C₅, Petersen, Hoffman-Singleton). Whether a degree-57 Moore graph exists is one of the most famous open problems in algebraic graph theory.", "difficulty_level_id": 5, "status": "open", "category_id": 3, "view_count": 223, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1414, "problem_number": "GRAPH-027", "title": "Barnette's Conjecture", "statement": "Does every cubic bipartite three-connected planar graph have a Hamiltonian cycle?", "background": "Barnette's conjecture proposes that a specific family of planar graphs—cubic (3-regular), bipartite, and 3-connected—always contains Hamiltonian cycles. This strengthens Tait's conjecture (disproven by counterexamples) by adding bipartiteness. Despite extensive computational verification and many partial results, no proof or counterexample is known. This is one of the most prominent open problems on Hamiltonian cycles.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 212, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1415, "problem_number": "GRAPH-028", "title": "Chvátal's Toughness Conjecture", "statement": "Is there a constant t such that every t-tough graph is Hamiltonian?", "background": "A graph is t-tough if removing any set S of vertices leaves at most |S|/t components. Chvátal conjectured that sufficiently tough graphs are Hamiltonian. Best known: every 2-tough graph on at least 3 vertices is Hamiltonian. But whether some finite t suffices in general is unknown. Toughness measures graph robustness; the conjecture would provide a simple sufficient condition for Hamiltonicity.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 178, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1416, "problem_number": "GRAPH-029", "title": "Cycle Double Cover Conjecture", "statement": "Does every bridgeless graph have a collection of cycles that covers each edge exactly twice?", "background": "The cycle double cover conjecture asserts that every bridgeless graph has a family of cycles where each edge appears in exactly two cycles. Equivalent formulations involve graph embeddings and flows. Despite being open since the 1970s, this elegant conjecture connects cycle structure, graph embeddings, and topological graph theory. Many restricted cases are proven, but the general case remains elusive.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 198, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1417, "problem_number": "GRAPH-030", "title": "Erdős-Gyárfás Conjecture", "statement": "Does every graph with minimum degree 3 contain cycles of lengths that are powers of 2?", "background": "Erdős and Gyárfás conjectured that cubic graphs (minimum degree 3) must contain cycles whose lengths are all distinct powers of 2. The best known result is that such graphs contain cycles of Ω(log log n) distinct even lengths. This problem connects extremal graph theory with additive combinatorics and the structure of cycle lengths in graphs.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 167, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1418, "problem_number": "GRAPH-031", "title": "Erdős-Hajnal Conjecture", "statement": "Does every graph family defined by a forbidden induced subgraph have polynomial-sized cliques or independent sets?", "background": "The Erdős-Hajnal conjecture states that for any graph H, there exists ε > 0 such that every H-free graph on n vertices contains a clique or independent set of size at least n^ε. This would be a dramatic strengthening of Ramsey theory, which only guarantees log-size structures. Proven for many specific H, but the general case is a central open problem in extremal combinatorics.", "difficulty_level_id": 5, "status": "open", "category_id": 3, "view_count": 234, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1419, "problem_number": "GRAPH-032", "title": "Linear Arboricity Conjecture", "statement": "Can every graph with maximum degree Δ be decomposed into at most ⌈(Δ+1)/2⌉ linear forests?", "background": "A linear forest is a disjoint union of paths. The linear arboricity conjecture states that graphs decompose into roughly Δ/2 linear forests. This would provide tight bounds on a natural graph decomposition parameter. Proven for many graph classes (planar graphs, graphs with large girth), but the general case remains open. Applications include edge coloring and bandwidth problems.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 156, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1420, "problem_number": "GRAPH-033", "title": "Lovász Conjecture", "statement": "Does every finite connected vertex-transitive graph contain a Hamiltonian path?", "background": "Lovász conjectured that vertex-transitive graphs (graphs looking the same from every vertex) always have Hamiltonian paths. Even stronger: do they have Hamiltonian cycles (except for K₂ and some Cayley graphs)? Known for many classes, but a general proof eludes us. This connects group theory, algebraic graph theory, and Hamiltonian paths. Named the \"Lovász Hamiltonian Path Problem.\"", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1421, "problem_number": "GRAPH-034", "title": "Oberwolfach Problem", "statement": "For which 2-regular graphs H can the complete graph be decomposed into edge-disjoint copies of H?", "background": "The Oberwolfach problem asks: given a 2-regular graph H (disjoint union of cycles), can K_n be decomposed into copies of H? This generalizes cycle decompositions and connects to the famous Oberwolfach conferences. Solutions are known for many cases (like single cycles), but a complete characterization remains open. This problem bridges graph decomposition with combinatorial designs.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 167, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1422, "problem_number": "GRAPH-035", "title": "Cubic Graph Pathwidth", "statement": "What is the maximum pathwidth of an n-vertex cubic graph?", "background": "Pathwidth measures how closely a graph resembles a path. For cubic (3-regular) graphs, the maximum pathwidth is conjectured to be around n/6, but exact bounds are unknown. This problem connects graph width parameters with regular graphs. Understanding pathwidth has implications for algorithms—many NP-hard problems become tractable on graphs of bounded pathwidth.", "difficulty_level_id": 3, "status": "open", "category_id": 3, "view_count": 134, "favorite_count": 10, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1423, "problem_number": "GRAPH-036", "title": "Snake-in-the-Box Problem", "statement": "What is the longest induced path in an n-dimensional hypercube graph?", "background": "A snake-in-the-box is a longest induced path in the n-dimensional hypercube Q_n. Known exact values for small n, but no formula for general n. This problem combines combinatorics, coding theory (Gray codes), and graph theory. Snakes have applications in error-correcting codes and analog-to-digital conversion. Finding optimal snakes remains computationally challenging as n grows.", "difficulty_level_id": 3, "status": "open", "category_id": 3, "view_count": 178, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1424, "problem_number": "GRAPH-037", "title": "Sumner's Conjecture", "statement": "Does every (2n-2)-vertex tournament contain every n-vertex oriented tree?", "background": "Sumner conjectured that tournaments (complete directed graphs) on 2n-2 vertices contain all oriented trees on n vertices as subgraphs. This would be a directed analogue of various tree embedding results. The best known bound is (4+o(1))n instead of 2n-2. This problem connects tournament theory with tree embeddings and Ramsey-type questions for directed graphs.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 156, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1425, "problem_number": "GRAPH-038", "title": "Tuza's Conjecture", "statement": "Can the edges of any graph be covered by at most 2ν triangles, where ν is the maximum size of a triangle packing?", "background": "Tuza conjectured that the minimum number of edges needed to hit all triangles is at most twice the maximum number of edge-disjoint triangles. This is a covering-packing duality question. Best known bound is 3ν. The conjecture would provide a tight relationship between triangle packings and triangle covers, with applications to approximation algorithms and combinatorial optimization.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1426, "problem_number": "GRAPH-039", "title": "Unfriendly Partition Conjecture", "statement": "Does every countable graph admit a partition where every vertex has at least as many neighbors outside its part as inside?", "background": "The unfriendly partition conjecture asks if vertices can be partitioned into two sets such that each vertex has at least as many \"unfriendly\" neighbors (in the other set) as \"friendly\" ones (in its own set). Proven for finite graphs, but open for countably infinite graphs. This problem combines graph theory with infinite combinatorics and has connections to social network models.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 145, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1427, "problem_number": "GRAPH-040", "title": "Zarankiewicz Problem", "statement": "What is the maximum number of edges in a bipartite graph on (m,n) vertices with no complete bipartite subgraph $K_{s,t}$?", "background": "The Zarankiewicz problem asks for ex(m,n;K_{s,t})—the maximum edges in an (m,n)-bipartite graph avoiding K_{s,t} as a subgraph. This is a fundamental problem in extremal graph theory, generalizing the Kővári–Sós–Turán theorem. Exact values are known for some parameters, but most cases remain open. Applications include incidence geometry and additive combinatorics.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 198, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1428, "problem_number": "GRAPH-041", "title": "Vizing's Conjecture", "statement": "For the Cartesian product of graphs $G \\square H$, is the domination number at least $\\gamma(G) \\cdot \\gamma(H)$?", "background": "Vizing conjectured that the domination number of the Cartesian product of two graphs is at least the product of their domination numbers. This would give a lower bound on how efficiently one can dominate product graphs. The conjecture has been verified for many special cases but remains open in general. It has connections to network design and distributed computing.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 172, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1429, "problem_number": "GRAPH-042", "title": "Hamiltonian Decomposition of Hypergraphs", "statement": "Do complete k-uniform hypergraphs admit Hamiltonian decompositions into tight cycles?", "background": "Walescki's theorem states that complete graphs have Hamiltonian decompositions. The hypergraph version asks whether complete k-uniform hypergraphs can be decomposed into tight Hamiltonian cycles. This is a natural generalization from graphs to hypergraphs, with connections to design theory and combinatorial structures.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 134, "favorite_count": 10, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1430, "problem_number": "GRAPH-043", "title": "Word-Representable Graphs: Letter Copies Bound", "statement": "Are there graphs on n vertices requiring more than floor(n/2) copies of each letter for word-representation?", "background": "Word-representable graphs can be encoded by words where two vertices are adjacent if their letters alternate in the word. The question asks whether any graph needs more than half the number of vertices as copies of each letter. This connects graph theory to formal languages and combinatorics on words.", "difficulty_level_id": 3, "status": "open", "category_id": 3, "view_count": 98, "favorite_count": 7, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1431, "problem_number": "GRAPH-044", "title": "Characterization of Word-Representable Planar Graphs", "statement": "Characterize which planar graphs are word-representable.", "background": "Word-representable graphs are those that can be encoded by words over their vertex set where adjacency corresponds to letter alternation. While some characterizations exist for special graph classes, characterizing word-representable planar graphs remains open. This combines planar graph structure with formal language properties.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 87, "favorite_count": 6, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1432, "problem_number": "GRAPH-045", "title": "Word-Representable Graphs: Forbidden Subgraph Characterization", "statement": "Characterize word-representable graphs in terms of forbidden induced subgraphs.", "background": "Many graph classes have elegant characterizations via forbidden subgraphs (e.g., planar graphs avoid K₅ and K₃,₃). The question asks for a similar characterization of word-representable graphs. Such a characterization would provide deep insight into the structure of these graphs and their connection to formal languages.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 92, "favorite_count": 7, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1433, "problem_number": "GRAPH-046", "title": "Word-Representable Near-Triangulations", "statement": "Characterize word-representable near-triangulations containing K₄.", "background": "Near-triangulations are planar graphs close to being triangulations. A characterization is known for K₄-free cases. The question asks to extend this to near-triangulations containing the complete graph K₄. This combines planar graph structure with word-representability constraints.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 76, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1434, "problem_number": "GRAPH-047", "title": "Representation Number 3 Classification", "statement": "Classify graphs with representation number exactly 3.", "background": "The representation number is the minimum number of letter copies needed to word-represent a graph. Graphs with representation number 1 and 2 are relatively well understood. The question asks for a complete classification of graphs requiring exactly 3 copies—not representable with 2, but possible with 3.", "difficulty_level_id": 3, "status": "open", "category_id": 3, "view_count": 81, "favorite_count": 6, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1435, "problem_number": "GRAPH-048", "title": "Crown Graphs and Longest Word-Representants", "statement": "Among bipartite graphs, do crown graphs require the longest word-representants?", "background": "Crown graphs are a specific family of bipartite graphs with a symmetric structure. The conjecture suggests they are extremal for word-representation length among bipartite graphs. This would identify which bipartite graphs are hardest to encode as words, with implications for the complexity of word-representation.", "difficulty_level_id": 3, "status": "open", "category_id": 3, "view_count": 73, "favorite_count": 5, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1436, "problem_number": "GRAPH-049", "title": "Line Graphs of Non-Word-Representable Graphs", "statement": "Is the line graph of a non-word-representable graph always non-word-representable?", "background": "The line graph operation transforms a graph into one where edges become vertices. The question asks whether word-non-representability is preserved under this operation. A positive answer would show that line graphs amplify the complexity of word-representation, while a counterexample would reveal subtle structural properties.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 84, "favorite_count": 6, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1437, "problem_number": "GRAPH-050", "title": "Translating Graph Problems to Word Problems", "statement": "Which hard graph problems can be efficiently solved by translating graphs to their word representations?", "background": "Word-representation provides an alternative encoding of graphs as strings over an alphabet. The question asks which computationally hard graph problems become tractable when working with word representations instead of adjacency lists or matrices. This could reveal new algorithmic techniques leveraging string algorithms and automata theory.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 105, "favorite_count": 8, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1438, "problem_number": "GRAPH-051", "title": "Imbalance Conjecture", "statement": "If every edge has imbalance ≥1, is the multiset of edge imbalances always graphic?", "background": "The imbalance of an edge is the absolute difference between the degrees of its endpoints. The conjecture asks whether the multiset of these imbalances can always realize a degree sequence of some graph when all imbalances are positive. This connects degree sequences with edge properties in a novel way.", "difficulty_level_id": 3, "status": "open", "category_id": 3, "view_count": 94, "favorite_count": 7, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1439, "problem_number": "GRAPH-052", "title": "Implicit Graph Conjecture", "statement": "Do slowly-growing hereditary graph families admit implicit representations?", "background": "The implicit graph conjecture concerns the existence of succinct encodings for hereditary families of graphs (closed under induced subgraphs) whose growth rate is subexponential. An implicit representation would allow efficient storage and adjacency queries. This has implications for data structures and graph databases.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 112, "favorite_count": 9, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1440, "problem_number": "GRAPH-053", "title": "Ryser's Conjecture", "statement": "For r-partite r-uniform hypergraphs, is the vertex cover number at most (r-1) times the matching number?", "background": "Ryser's conjecture relates the minimum transversal (vertex cover) size to maximum matching size in hypergraphs. For graphs (r=2) this is König's theorem. The conjecture proposes a tight bound for hypergraphs: τ ≤ (r-1)ν. This is a central open problem in hypergraph theory with connections to combinatorial optimization.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 156, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1441, "problem_number": "GRAPH-054", "title": "Second Neighborhood Problem", "statement": "Does every oriented graph have a vertex with at least as many vertices at distance 2 as at distance 1?", "background": "The second neighborhood problem asks whether oriented graphs always contain a vertex whose second neighborhood (vertices at distance exactly 2) is at least as large as its first neighborhood (out-neighbors). This has been conjectured by several researchers and has connections to tournament theory and Seymour's second neighborhood conjecture.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 128, "favorite_count": 10, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1442, "problem_number": "GRAPH-055", "title": "Teschner's Bondage Number Conjecture", "statement": "Is the bondage number of a graph always ≤ 3Δ/2, where Δ is the maximum degree?", "background": "The bondage number is the minimum number of edges whose removal increases the domination number. Teschner conjectured an upper bound of 3Δ/2 in terms of maximum degree Δ. This would establish a fundamental relationship between edge removal sensitivity and local graph structure in domination problems.", "difficulty_level_id": 3, "status": "open", "category_id": 3, "view_count": 89, "favorite_count": 7, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1443, "problem_number": "GRAPH-056", "title": "Tutte's 5-Flow Conjecture", "statement": "Does every bridgeless graph have a nowhere-zero 5-flow?", "background": "Tutte's 5-flow conjecture is one of the most famous problems in graph theory. A nowhere-zero k-flow is an orientation and edge-labeling with values in {±1,...,±(k-1)} satisfying flow conservation. The conjecture states that 5 colors suffice for all bridgeless graphs. Related to the four-color theorem and still wide open despite much research.", "difficulty_level_id": 5, "status": "open", "category_id": 3, "view_count": 267, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1444, "problem_number": "GRAPH-057", "title": "Tutte's 4-Flow Conjecture for Petersen-Minor-Free Graphs", "statement": "Does every Petersen-minor-free bridgeless graph have a nowhere-zero 4-flow?", "background": "This is a refinement of Tutte's 5-flow conjecture for graphs without Petersen graph minors. The Petersen graph is known to require 5 colors for nowhere-zero flows, so excluding it might allow 4-flows. This conjecture connects graph minors, nowhere-zero flows, and the special role of the Petersen graph in combinatorics.", "difficulty_level_id": 5, "status": "open", "category_id": 3, "view_count": 198, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1445, "problem_number": "GRAPH-058", "title": "Woodall's Conjecture", "statement": "Is the minimum dicut size equal to the maximum number of disjoint dijoins in a directed graph?", "background": "Woodall's conjecture is a directed graph analogue of Menger's theorem. A dicut is a set of arcs whose removal disconnects the graph directionally, and a dijoin connects specified vertex pairs. The conjecture proposes a min-max relation, which would be a fundamental packing-covering duality for directed graphs.", "difficulty_level_id": 4, "status": "open", "category_id": 3, "view_count": 134, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1446, "problem_number": "ALG-001", "title": "Birch-Tate Conjecture", "statement": "Relate the order of the center of the Steinberg group of the ring of integers to the Dedekind zeta function.", "background": "The Birch-Tate conjecture connects algebraic K-theory to special values of zeta functions. It predicts a precise relationship between the center of the Steinberg group St(O_K) of a number field K and the value of its Dedekind zeta function at s=-1. This is a fundamental connection between algebra and analytic number theory.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 187, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1447, "problem_number": "ALG-002", "title": "Casas-Alvero Conjecture", "statement": "If a polynomial of degree d over a field of characteristic 0 shares a factor with each of its first d-1 derivatives, must it be $(x-a)^d$?", "background": "The Casas-Alvero conjecture states that a polynomial sharing roots with all its derivatives (up to degree d-1) must be a power of a linear polynomial. Despite its elementary statement, it remains open. The conjecture has been verified for many special cases but lacks a general proof.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 203, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1448, "problem_number": "ALG-003", "title": "Connes Embedding Problem", "statement": "Can every finite von Neumann algebra be embedded into an ultrapower of the hyperfinite II₁ factor?", "background": "The Connes embedding problem is a central question in operator algebra theory. It asks whether all separable II₁ factors embed into the ultrapower of the hyperfinite II₁ factor. This problem connects functional analysis, quantum information theory, and logic. Recent claimed solutions using quantum computing have generated significant interest.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 289, "favorite_count": 22, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1449, "problem_number": "ALG-004", "title": "Crouzeix's Conjecture", "statement": "Is $\\|f(A)\\| \\leq 2 \\sup_{z \\in W(A)} |f(z)|$ for any matrix A and analytic function f on the numerical range W(A)?", "background": "Crouzeix's conjecture bounds the matrix norm of f(A) by twice the supremum of |f| over the numerical range of A. The constant 2 would be optimal. This conjecture connects matrix theory, complex analysis, and numerical analysis. The best known bound is approximately 11.08, far from the conjectured 2.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 156, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1450, "problem_number": "ALG-005", "title": "Determinantal Conjecture", "statement": "Characterize the determinant of the sum of two normal matrices.", "background": "The determinantal conjecture seeks inequalities or characterizations for det(A+B) when A and B are normal matrices. While det(AB) = det(A)det(B) is well known, the sum of normal matrices presents challenges. This problem connects linear algebra with operator theory and has applications in quantum mechanics.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 134, "favorite_count": 10, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1451, "problem_number": "ALG-006", "title": "Eilenberg-Ganea Conjecture", "statement": "Does every group with cohomological dimension 2 have a 2-dimensional Eilenberg-MacLane space K(G,1)?", "background": "The Eilenberg-Ganea conjecture asks whether cohomological dimension equals geometric dimension for groups. Specifically, if cd(G)=2, does there exist a 2-dimensional CW complex with fundamental group G? The conjecture is known to hold for cd ≠ 2. This connects algebraic topology with group theory.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 178, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1452, "problem_number": "ALG-007", "title": "Farrell-Jones Conjecture", "statement": "Are the assembly maps in algebraic K-theory and L-theory isomorphisms?", "background": "The Farrell-Jones conjecture predicts that certain assembly maps are isomorphisms for all groups. This would have major consequences for the computation of algebraic K-theory and L-theory groups. The conjecture has been verified for many important classes of groups including hyperbolic groups and arithmetic groups.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 165, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1453, "problem_number": "ALG-008", "title": "Finite Lattice Representation Problem", "statement": "Is every finite lattice isomorphic to the congruence lattice of some finite algebra?", "background": "The finite lattice representation problem asks whether every finite lattice can be realized as the congruence lattice of a finite algebra. While every finite lattice is the congruence lattice of some algebra, requiring finiteness of the algebra is much more restrictive. This is a central problem in universal algebra.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 142, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1454, "problem_number": "ALG-009", "title": "Hadamard Matrix Conjecture", "statement": "Does a Hadamard matrix of order 4k exist for every positive integer k?", "background": "The Hadamard conjecture states that Hadamard matrices (square matrices with entries ±1 and mutually orthogonal rows) exist for all orders divisible by 4. These matrices have applications in coding theory, cryptography, and experimental design. The smallest open case is k=167 (order 668).", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 245, "favorite_count": 19, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1455, "problem_number": "ALG-010", "title": "Köthe Conjecture", "statement": "If a ring has no nil two-sided ideal besides {0}, does it also have no nil one-sided ideal besides {0}?", "background": "The Köthe conjecture asks whether the absence of nontrivial nil ideals implies the absence of nontrivial nil one-sided ideals. A nil ideal is one where every element is nilpotent. This has been a central problem in ring theory for decades, with connections to the structure theory of noncommutative rings.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 167, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1456, "problem_number": "ALG-011", "title": "Perfect Cuboid", "statement": "Does there exist a perfect cuboid—a rectangular parallelepiped with integer edges, face diagonals, and space diagonal?", "background": "A perfect cuboid would be a box where all edges, face diagonals, and the space diagonal are integers. Despite extensive computational searches, no perfect cuboid has been found, nor has non-existence been proven. This is a Diophantine problem with connections to number theory and geometry.", "difficulty_level_id": 3, "status": "open", "category_id": 4, "view_count": 312, "favorite_count": 24, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1457, "problem_number": "ALG-012", "title": "Rota's Basis Conjecture", "statement": "Given n bases of an n-dimensional matroid, can we find n disjoint rainbow bases?", "background": "Rota's basis conjecture asks whether n disjoint bases B₁,...,Bₙ of a matroid of rank n can be rearranged into an n×n matrix where each row is a basis and each column is a transversal (rainbow basis). This elegant conjecture connects matroid theory with combinatorics and has resisted many attempts at proof.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1458, "problem_number": "MOD-001", "title": "Cherlin-Zilber Conjecture", "statement": "Is every simple group with a stable first-order theory an algebraic group over an algebraically closed field?", "background": "The Cherlin-Zilber conjecture (also called the algebraicity conjecture) proposes that infinite simple groups with stable theories are essentially algebraic groups. This would classify a vast class of model-theoretically tame groups. The conjecture connects model theory, group theory, and algebraic geometry in a profound way.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 176, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1459, "problem_number": "MOD-002", "title": "Generalized Star Height Problem", "statement": "Can all regular languages be expressed with generalized regular expressions having bounded star height?", "background": "The generalized star height problem asks whether there's a universal bound on the nesting depth of Kleene stars needed to express regular languages. This is a fundamental question in formal language theory and automata theory, with connections to computational complexity and logic.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 143, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1460, "problem_number": "MOD-003", "title": "Hilbert's Tenth Problem for Number Fields", "statement": "For which number fields is there an algorithm to determine if a Diophantine equation has solutions?", "background": "Hilbert's tenth problem asked for an algorithm to solve Diophantine equations over the integers—proven impossible by Matiyasevich. The question for other number fields remains open. It's known to be undecidable for some fields and decidable for others. Determining exactly which fields admit such algorithms is a major open problem.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 234, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1461, "problem_number": "MOD-004", "title": "Vaught Conjecture", "statement": "Does every complete first-order theory in a countable language have countably many, $\\aleph_0$, or $2^{\\aleph_0}$ countable models?", "background": "Vaught's conjecture states that the number of countable models of a complete theory is either finite, countably infinite, or continuum. This would rule out intermediate cardinalities. The conjecture connects model theory with descriptive set theory and has deep connections to the structure of mathematical logic.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 198, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1462, "problem_number": "MOD-005", "title": "Tarski's Exponential Function Problem", "statement": "Is the theory of the real numbers with addition, multiplication, and exponentiation decidable?", "background": "Tarski proved that the theory of real closed fields is decidable. Adding exponentiation makes the question much harder. Decidability would mean an algorithm exists to determine truth of statements involving exp. This has implications for automated theorem proving and connections to transcendental number theory.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 256, "favorite_count": 20, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1463, "problem_number": "MOD-006", "title": "Stable Field Conjecture", "statement": "Is every infinite field with a stable first-order theory separably closed?", "background": "The stable field conjecture predicts that infinite fields with stable theories are separably closed. Stable theories are model-theoretically well-behaved. This conjecture would classify all stable fields, providing a complete understanding of these algebraically important structures through a model-theoretic lens.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 167, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1464, "problem_number": "MOD-007", "title": "Henson Graphs Finite Model Property", "statement": "Do Henson graphs have the finite model property?", "background": "Henson graphs are universal homogeneous graphs omitting certain finite subgraphs. The finite model property asks whether every satisfiable sentence has a finite model. This question connects infinite graph theory, model theory, and combinatorics, with implications for the decidability of their first-order theories.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 123, "favorite_count": 9, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1465, "problem_number": "MOD-008", "title": "O-Minimal Theory with Trans-Exponential Growth", "statement": "Does there exist an o-minimal first-order theory with a trans-exponential (rapid growth) function?", "background": "O-minimal structures are ordered structures where definable sets have simple topology. Known o-minimal structures include real closed fields and structures with restricted analytic functions. The question asks whether o-minimality is compatible with very fast-growing functions, testing the limits of tame model theory.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 145, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1466, "problem_number": "MOD-009", "title": "Infinite Minimal Field Algebraic Closure", "statement": "Is every infinite minimal field of characteristic zero algebraically closed?", "background": "A minimal structure is one where every definable subset is finite or cofinite. The question asks whether infinite fields with this property must be algebraically closed (when char=0). This would characterize the simplest infinite fields from a model-theoretic perspective, connecting field theory with minimality.", "difficulty_level_id": 4, "status": "open", "category_id": 4, "view_count": 134, "favorite_count": 10, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1467, "problem_number": "MOD-010", "title": "Keisler's Order", "statement": "Determine the structure of Keisler's order on first-order theories.", "background": "Keisler's order compares first-order theories based on the complexity of their ultrapowers. Understanding this order would classify theories by their model-theoretic complexity. Recent breakthroughs have shed light on the order's structure, but a complete classification remains elusive. This connects with classification theory and stability.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 156, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1468, "problem_number": "ALG-013", "title": "Serre's Conjecture II", "statement": "For simply connected semisimple algebraic groups over fields of cohomological dimension ≤2, is $H^1(F,G) = 0$?", "background": "Serre's Conjecture II predicts that the first Galois cohomology of simply connected semisimple groups vanishes over fields of small cohomological dimension. This would have major implications for the classification of algebraic groups and forms. The conjecture is known for various classes of fields but remains open in general.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 178, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1469, "problem_number": "ALG-014", "title": "Serre's Positivity Conjecture", "statement": "If R is a regular local ring and P,Q are prime ideals with $\\dim(R/P) + \\dim(R/Q) = \\dim(R)$, is $\\chi(R/P, R/Q) > 0$?", "background": "Serre's positivity conjecture predicts that the Euler characteristic (intersection multiplicity) is positive when dimensions add correctly. This is part of a broader set of homological conjectures in commutative algebra. The conjecture would provide fundamental information about the structure of modules over regular rings.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 145, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1470, "problem_number": "ALG-015", "title": "Uniform Boundedness Conjecture for Rational Points", "statement": "Is there a bound N(g,d) such that all curves of genus g≥2 over degree d number fields have at most N(g,d) rational points?", "background": "The uniform boundedness conjecture asks whether the number of rational points on curves of genus ≥2 is uniformly bounded in terms of genus and field degree. This would be a remarkable strengthening of Faltings' theorem (finite number of points). The conjecture connects arithmetic geometry with Diophantine equations.", "difficulty_level_id": 5, "status": "open", "category_id": 4, "view_count": 213, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1479, "problem_number": "TOP-001", "title": "Baum-Connes Conjecture", "statement": "Is the assembly map in K-theory an isomorphism for all locally compact groups?", "background": "The Baum-Connes conjecture predicts that a certain assembly map from equivariant K-homology to the K-theory of group C*-algebras is an isomorphism. This would have major consequences for the Novikov conjecture, index theory, and the structure of operator algebras. Known for many groups, general case open.", "difficulty_level_id": 5, "status": "open", "category_id": 7, "view_count": 198, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1480, "problem_number": "TOP-002", "title": "Berge Conjecture", "statement": "Are Berge knots the only knots in S³ admitting lens space surgeries?", "background": "The Berge conjecture states that Berge knots (constructed via a specific procedure) are the only knots in the 3-sphere that admit Dehn surgeries yielding lens spaces. This would classify all such knots, providing deep insight into the relationship between knot theory and 3-manifold topology.", "difficulty_level_id": 4, "status": "open", "category_id": 7, "view_count": 167, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1481, "problem_number": "TOP-003", "title": "Borel Conjecture", "statement": "Are aspherical closed manifolds determined up to homeomorphism by their fundamental groups?", "background": "The Borel conjecture predicts that aspherical closed manifolds (those with contractible universal cover) are rigid—completely determined by their fundamental group up to homeomorphism. This would be a remarkable topological rigidity result, currently known only for special classes of manifolds.", "difficulty_level_id": 5, "status": "open", "category_id": 7, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1482, "problem_number": "TOP-004", "title": "Hilbert-Smith Conjecture", "statement": "If a locally compact group acts faithfully and continuously on a manifold, must it be a Lie group?", "background": "The Hilbert-Smith conjecture asks whether every locally compact group with a continuous faithful action on a manifold is necessarily a Lie group. This would rule out p-adic groups acting on manifolds, resolving a fundamental question about the symmetries of topological spaces.", "difficulty_level_id": 5, "status": "open", "category_id": 7, "view_count": 212, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1483, "problem_number": "TOP-005", "title": "Novikov Conjecture", "statement": "Are certain polynomials in Pontryagin classes homotopy invariants?", "background": "The Novikov conjecture states that higher signatures (certain rational combinations of Pontryagin numbers) are oriented homotopy invariants. This has profound consequences for manifold topology, surgery theory, and K-theory. Proven for many classes of groups, but the general case remains open.", "difficulty_level_id": 5, "status": "open", "category_id": 7, "view_count": 234, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1484, "problem_number": "TOP-006", "title": "Unknotting Problem", "statement": "Can unknots be recognized in polynomial time?", "background": "The unknotting problem asks whether there exists a polynomial-time algorithm to determine if a knot diagram represents the unknot. While algorithms exist (exponential time), polynomial-time decidability remains open. This is a central problem in computational topology with connections to complexity theory.", "difficulty_level_id": 4, "status": "open", "category_id": 7, "view_count": 256, "favorite_count": 20, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1485, "problem_number": "TOP-007", "title": "Volume Conjecture", "statement": "Do quantum invariants of knots determine their hyperbolic volume?", "background": "The volume conjecture predicts an exponential relationship between the colored Jones polynomial (a quantum invariant) and the hyperbolic volume of a knot complement. This would connect quantum topology with hyperbolic geometry in a striking way, revealing deep structures in 3-dimensional topology.", "difficulty_level_id": 5, "status": "open", "category_id": 7, "view_count": 201, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1486, "problem_number": "TOP-008", "title": "Whitehead Conjecture", "statement": "Is every connected subcomplex of a 2-dimensional aspherical CW complex also aspherical?", "background": "The Whitehead conjecture asks whether asphericity (having contractible universal cover) is preserved under taking subcomplexes in dimension 2. This would clarify the local structure of aspherical spaces and has connections to group theory and low-dimensional topology.", "difficulty_level_id": 4, "status": "open", "category_id": 7, "view_count": 143, "favorite_count": 11, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1487, "problem_number": "TOP-009", "title": "Zeeman Conjecture", "statement": "Is $K \\times [0,1]$ collapsible for every finite contractible 2-dimensional CW complex K?", "background": "The Zeeman conjecture predicts that the product of any finite contractible 2-complex with an interval is collapsible (can be reduced to a point by elementary collapses). This relates to the Poincaré conjecture and questions about higher-dimensional manifolds. A counterexample would have major implications.", "difficulty_level_id": 4, "status": "open", "category_id": 7, "view_count": 134, "favorite_count": 10, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1488, "problem_number": "COMB-001", "title": "1/3-2/3 Conjecture", "statement": "Does every non-total finite poset have two elements x,y with P(x before y in random linear extension) ∈ [1/3, 2/3]?", "background": "The 1/3-2/3 conjecture asks whether finite partially ordered sets (not totally ordered) always contain a pair with intermediate probability of appearing in a certain order. This connects order theory with probability and has implications for sorting algorithms and social choice theory.", "difficulty_level_id": 3, "status": "open", "category_id": 2, "view_count": 124, "favorite_count": 9, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1489, "problem_number": "COMB-002", "title": "Lonely Runner Conjecture", "statement": "If k runners with distinct speeds run on a unit circle, will each runner be \"lonely\" (≥1/k away from others) at some time?", "background": "The lonely runner conjecture predicts that in a system of runners with different speeds on a circular track, each runner will at some point be far from all others. Verified for k≤7, this problem connects view obstruction, Diophantine approximation, and number theory.", "difficulty_level_id": 4, "status": "open", "category_id": 2, "view_count": 156, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1490, "problem_number": "COMB-003", "title": "Sunflower Conjecture", "statement": "Can the minimum size for sunflowers be bounded by an exponential (not super-exponential) function of k?", "background": "The sunflower conjecture asks whether families of k-element sets containing a sunflower (r sets with common \"core\") require only exponentially many sets in k. Recent progress by Alweiss et al. improved bounds but the original conjecture remains open. Fundamental for extremal combinatorics.", "difficulty_level_id": 4, "status": "open", "category_id": 2, "view_count": 178, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1491, "problem_number": "COMB-004", "title": "Union-Closed Sets Conjecture", "statement": "For any finite union-closed family of sets, does some element appear in at least half the sets?", "background": "Frankl's union-closed sets conjecture (also called the union-closed set conjecture) states that in any family of sets closed under unions, at least one element appears in ≥50% of the sets. Despite its elementary statement, this problem has resisted all attempts at proof.", "difficulty_level_id": 4, "status": "open", "category_id": 2, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1492, "problem_number": "COMB-005", "title": "Ramsey Number R(5,5)", "statement": "What is the exact value of the Ramsey number R(5,5)?", "background": "Ramsey theory asks: in any 2-coloring of edges of the complete graph Kₙ, what's the minimum n guaranteeing a monochromatic K₅? Known: 43 ≤ R(5,5) ≤ 48. Finding the exact value would be a major breakthrough. Paul Erdős famously said R(6,6) would require alien technology.", "difficulty_level_id": 4, "status": "open", "category_id": 2, "view_count": 267, "favorite_count": 21, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1495, "problem_number": "NUM-001", "title": "Singmaster's Conjecture", "statement": "Is there a finite upper bound on multiplicities of entries >1 in Pascal's triangle?", "background": "Singmaster's conjecture asks whether any number (other than 1) appears in Pascal's triangle only finitely many times. Known: no entry appears more than 8 times. A proof would reveal deep structure in binomial coefficients and their divisibility properties.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 178, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1496, "problem_number": "NUM-002", "title": "Odd Perfect Numbers", "statement": "Do any odd perfect numbers exist?", "background": "A perfect number equals the sum of its proper divisors. All known perfect numbers are even (form 2^(p-1)(2^p-1) for Mersenne primes). Whether odd perfect numbers exist is one of the oldest open problems in mathematics, dating to ancient Greece. If they exist, they must be very large (>10^1500).", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 412, "favorite_count": 32, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1497, "problem_number": "NUM-003", "title": "Infinitude of Perfect Numbers", "statement": "Are there infinitely many perfect numbers?", "background": "All known perfect numbers are even and correspond to Mersenne primes via Euclid-Euler theorem. The question reduces to: are there infinitely many Mersenne primes? This remains open despite extensive computational searches. Connected to the distribution of primes and special number forms.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 345, "favorite_count": 27, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1498, "problem_number": "NUM-004", "title": "Quasiperfect Numbers", "statement": "Do quasiperfect numbers exist?", "background": "A quasiperfect number n has σ(n) = 2n+1 (sum of divisors is one more than twice the number). No quasiperfect numbers are known. If they exist, they must be odd perfect squares >10^35. This problem connects divisor functions with perfect number theory.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 167, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1499, "problem_number": "NUM-005", "title": "Lychrel Numbers", "statement": "Do Lychrel numbers exist in base 10?", "background": "A Lychrel number never forms a palindrome through iterative reverse-and-add process. 196 is the first candidate—after billions of iterations, no palindrome found. Proving existence or non-existence would resolve this computational mystery connecting palindromes with iteration dynamics.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 234, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1500, "problem_number": "NUM-006", "title": "Odd Weird Numbers", "statement": "Do odd weird numbers exist?", "background": "Weird numbers are abundant but not semiperfect (no subset of divisors sums to the number). All known weird numbers are even. Finding an odd weird number or proving none exist would reveal deep structure in additive properties of divisors.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1501, "problem_number": "NUM-007", "title": "Infinitude of Amicable Pairs", "statement": "Are there infinitely many pairs of amicable numbers?", "background": "Amicable pairs (m,n) satisfy σ(m)-m=n and σ(n)-n=m. Over 12 million pairs known, but infinity unproven. Related to perfect numbers and sociable chains. Erdős-Rieger heuristics suggest infinity, but proof remains elusive.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 212, "favorite_count": 17, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1502, "problem_number": "NUM-008", "title": "Pi Normality", "statement": "Is π a normal number (all digits equally frequent in all bases)?", "background": "A normal number has each digit appearing with equal asymptotic frequency in every base. While π appears statistically normal (verified to trillions of digits), no proof exists. This connects transcendental numbers, digit distribution, and randomness in mathematical constants.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 389, "favorite_count": 30, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1503, "problem_number": "NUM-009", "title": "Algebraic Number Normality", "statement": "Are all irrational algebraic numbers normal?", "background": "The question asks whether every irrational root of a polynomial with integer coefficients has all digits equally distributed in every base. A positive answer would be a remarkable connection between algebraic structure and digit statistics. Currently, we cannot prove normality for any specific algebraic irrational.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 201, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1504, "problem_number": "NUM-010", "title": "Gilbreath's Conjecture", "statement": "Does iterating unsigned differences on prime sequence always yield 1 as first element?", "background": "Start with primes 2,3,5,7,11,... Take absolute differences: 1,2,2,4,... Repeat. Conjecture: first element is always 1. Verified to huge primes, but unproven. This reveals hidden regularity in prime gaps with implications for prime distribution.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 156, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1505, "problem_number": "NUM-011", "title": "Lander-Parkin-Selfridge Conjecture", "statement": "If Σᵢ aᵢᵏ = Σⱼ bⱼᵏ with m terms on left, n on right, is m+n ≥ k?", "background": "The LPS conjecture generalizes Fermat's Last Theorem to sums of k-th powers. It predicts you need at least k terms total for nontrivial solutions. Counterexamples exist for specific cases, but the general conjecture remains open with implications for Diophantine equations.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 178, "favorite_count": 14, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1506, "problem_number": "NUM-012", "title": "Class Number Problem", "statement": "Are there infinitely many real quadratic fields with class number 1 (unique factorization)?", "background": "The class number problem asks whether infinitely many real quadratic number fields Q(√d) have unique factorization. For imaginary quadratic fields, Heegner-Baker-Stark proved only finitely many exist. The real case remains open—a fundamental question in algebraic number theory.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 198, "favorite_count": 16, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1507, "problem_number": "NUM-013", "title": "Hilbert's 12th Problem", "statement": "Extend Kronecker-Weber theorem to abelian extensions of arbitrary number fields.", "background": "Hilbert's 12th problem asks for explicit construction of abelian extensions of number fields via special values of transcendental functions (generalizing cyclotomic fields for Q). Partial progress via complex multiplication, but general case remains one of Hilbert's unsolved problems.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 187, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1508, "problem_number": "NUM-014", "title": "Leopoldt's Conjecture", "statement": "Does the p-adic regulator of an algebraic number field never vanish?", "background": "Leopoldt's conjecture predicts that the p-adic regulator (a p-adic analogue of the classical regulator from Dirichlet's unit theorem) is always nonzero. This has major implications for Iwasawa theory and the structure of p-adic L-functions.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 156, "favorite_count": 12, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1509, "problem_number": "NUM-015", "title": "Siegel Zeros", "statement": "Do Siegel zeros (real zeros of Dirichlet L-functions near s=1) exist?", "background": "Siegel zeros are hypothetical exceptional real zeros of L-functions very close to s=1. If they exist, they violate the Generalized Riemann Hypothesis. Their existence would have major consequences for prime distribution in arithmetic progressions. Most believe they don't exist.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 234, "favorite_count": 18, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1510, "problem_number": "NUM-016", "title": "Schanuel's Conjecture", "statement": "For e and π: are they algebraically independent? Is e+π, eπ, π^e, etc. transcendental?", "background": "Schanuel's conjecture is a fundamental statement about transcendence degrees. It implies e and π are algebraically independent and that expressions like e+π, eπ, π^π are transcendental. Proving it would resolve many open questions in transcendental number theory at once.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 287, "favorite_count": 22, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1511, "problem_number": "NUM-017", "title": "Euler-Mascheroni Constant Irrationality", "statement": "Is the Euler-Mascheroni constant γ irrational? Transcendental?", "background": "The Euler-Mascheroni constant γ ≈ 0.5772 appears throughout analysis and number theory. We don't even know if it's irrational! Proving irrationality or transcendence would be a major achievement. Related constants like Catalan's G and ζ(3) face similar questions.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 323, "favorite_count": 25, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1512, "problem_number": "NUM-018", "title": "Littlewood Conjecture", "statement": "For any α,β ∈ ℝ, is lim inf_{n→∞} n·||nα||·||nβ|| = 0?", "background": "Littlewood's conjecture connects Diophantine approximation of pairs of real numbers. It predicts that for any two reals, you can simultaneously approximate both well infinitely often. Related to continued fractions and dynamics on homogeneous spaces. Proved for many special cases.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 189, "favorite_count": 15, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1513, "problem_number": "NUM-019", "title": "Four Exponentials Conjecture", "statement": "If x₁,x₂ and y₁,y₂ are linearly independent over ℚ, is at least one of e^(xᵢyⱼ) transcendental?", "background": "The four exponentials conjecture states that you can't have all four values e^(x₁y₁), e^(x₁y₂), e^(x₂y₁), e^(x₂y₂) algebraic when the xᵢ and yⱼ satisfy independence conditions. Weaker than Schanuel's conjecture but still wide open. Six exponentials theorem is the proven weaker version.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 167, "favorite_count": 13, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1514, "problem_number": "NUM-020", "title": "Integer Factorization Polynomial Time", "statement": "Can integer factorization be done in polynomial time?", "background": "The integer factorization problem asks whether factoring large integers into primes can be done efficiently (polynomial time). RSA cryptography relies on it being hard. Shor's algorithm solves it on quantum computers, but classical complexity remains unknown. Related to P vs NP.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 456, "favorite_count": 35, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1515, "problem_number": "PDE-001", "title": "Navier-Stokes Existence and Smoothness", "statement": "Do smooth solutions to Navier-Stokes equations exist globally in 3D? Or do finite-time singularities occur?", "background": "The Navier-Stokes existence and smoothness problem is one of the seven Millennium Prize Problems. It asks whether smooth solutions to the 3D Navier-Stokes equations exist for all time, or whether finite-time blow-up can occur. Fundamental for fluid dynamics and mathematical physics.", "difficulty_level_id": 5, "status": "open", "category_id": 9, "view_count": 512, "favorite_count": 39, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set_id": 2 }, { "id": 1516, "problem_number": "GEOM-001", "title": "Sphere Packing Problem Higher Dimensions", "statement": "What is the optimal sphere packing density in dimensions >3?", "background": "The sphere packing problem asks for the densest way to pack spheres in n-dimensional space. Solved in dimensions 1,2,3 (Kepler's conjecture, proved by Hales), 8, and 24 (Viazovska). Dimensions 4-7 and ≥9 remain open. Connections to lattices, coding theory, and optimization.", "difficulty_level_id": 5, "status": "open", "category_id": 6, "view_count": 298, "favorite_count": 23, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null }, { "id": 1517, "problem_number": "HL-A", "title": "Hardy-Littlewood Conjecture A (Prime k-tuples)", "statement": "Let $a_1, \\ldots, a_k$ be given integers. Then there exist infinitely many positive integers $n$ such that $n + a_1, \\ldots, n + a_k$ are all prime, provided that for every prime $p$, there exists an integer $m$ such that $(m + a_i, p) = 1$ for all $i$.", "background": "The first Hardy-Littlewood conjecture, also known as the prime k-tuples conjecture, generalizes the twin prime conjecture. It states that the asymptotic frequency of any admissible prime constellation can be computed explicitly. The case $k=2$ with $(a_1, a_2) = (0, 2)$ is the twin prime conjecture. Yitang Zhang proved in 2013 that there exists at least one 2-tuple with gap ≤70,000,000 (later improved to 246) that appears infinitely often. The full conjecture remains open and is considered one of the most important unsolved problems in number theory.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set_id": 7 }, { "id": 1518, "problem_number": "HL-B", "title": "Hardy-Littlewood Conjecture B (Second Conjecture)", "statement": "For all integers $x, y \\geq 2$, we have $\\pi(x+y) \\leq \\pi(x) + \\pi(y)$, where $\\pi(n)$ denotes the prime counting function (the number of primes less than or equal to $n$).", "background": "The second Hardy-Littlewood conjecture states the subadditivity of the prime counting function. In 1974, Hensley and Richards proved that Conjecture A and Conjecture B are incompatible with each other - they cannot both be true. Since Conjecture A (the prime k-tuples conjecture) is considered more likely to be true based on computational evidence and its connections to the twin prime conjecture, most number theorists believe Conjecture B is actually false, despite appearing plausible. This represents a fascinating case where intuitive conjectures can contradict each other.", "difficulty_level_id": 5, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set_id": 7 }, { "id": 1519, "problem_number": "HL-F", "title": "Hardy-Littlewood Conjecture F (Primes in Quadratic Polynomials)", "statement": "For a polynomial $f(x) = ax^2 + bx + c$ with $a > 0$, $\\gcd(a,b,c) = 1$, and discriminant $\\Delta = b^2 - 4ac$ not a perfect square, the polynomial takes infinitely many prime values. Furthermore, the number $P(n)$ of primes of the form $f(x) \\leq n$ satisfies an asymptotic formula $P(n) \\sim A \\cdot \\frac{\\sqrt{n}}{\\log n}$ where $A$ depends on $a, b, c$ but not on $n$.", "background": "Conjecture F is a special case of the Bateman-Horn conjecture and concerns primes represented by quadratic polynomials. It predicts not only the infinitude of such primes but also their asymptotic density. The constant A can take values larger or smaller than 1, meaning some polynomials are especially rich in primes while others are especially poor. For example, $4x^2 - 2x + 41$ has $A \\approx 6.6$, making it nearly 7 times as likely to produce primes as random numbers of the same size. This conjecture explains the visible patterns in the Ulam spiral. Despite extensive computational verification, no polynomial has been proven to produce infinitely many primes except linear polynomials (Dirichlet's theorem).", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set_id": 7 }, { "id": 1855, "problem_number": "GUY-A4", "title": "The Prime Number Race", "statement": "Let $\\pi(n; a, b)$ be the number of primes $p \\le n$ with $p \\equiv a \\pmod b$. For every $a$ and $b$ with $a \\perp b$, are there infinitely many values of $n$ for which $\\pi(n; a, b) > \\pi(n; a_1, b)$ for every $a_1 \\not\\equiv a \\pmod b$?", "background": "Turán was particularly interested in the prime number race. Knapowski & Turán settled special cases, but the general problem is wide open. Chebyshev noted that $\\pi(n; 1, 3) < \\pi(n; 2, 3)$ for small values of $n$, but this inequality is reversed for very large $n$. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A4.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set_id": 9 }, { "id": 1857, "problem_number": "GUY-A5b", "title": "Erdős $3000 Conjecture on Arithmetic Progressions", "statement": "Let $\\{a_i\\}$ be any infinite sequence of integers for which $\\sum 1/a_i$ is divergent. Does the sequence contain arbitrarily long arithmetic progressions?", "background": "Erdős offered $3000.00 for a proof or disproof of this conjecture. This is a generalization of the arithmetic progressions of primes problem. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A5.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set_id": 9 }, { "id": 1858, "problem_number": "GUY-A6", "title": "Consecutive Primes in Arithmetic Progression", "statement": "Are there arbitrarily long arithmetic progressions of consecutive primes? That is, for any positive integer $k$, do there exist $k$ consecutive primes $p_n, p_{n+1}, \\ldots, p_{n+k-1}$ in arithmetic progression?", "background": "Known examples include the 4-term sequences 251, 257, 263, 269 and 1741, 1747, 1753, 1759. Dubner, Forbes, Lygeros, Mizony & Zimmermann found 10 consecutive primes in arithmetic progression in 1998. It is not known if there are infinitely many sets of three consecutive primes in arithmetic progression. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A6.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set_id": 9 }, { "id": 1859, "problem_number": "GUY-A7a", "title": "Infinitude of Sophie Germain Primes", "statement": "Are there infinitely many Sophie Germain primes? A prime $p$ is called a Sophie Germain prime if $2p + 1$ is also prime.", "background": "It is believed, but not known, that there are infinitely many Sophie Germain primes. Dubner has found many large examples. The largest known Sophie Germain prime has over 24000 decimal digits. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A7.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set_id": 9 }, { "id": 1860, "problem_number": "GUY-A7b", "title": "Shanks Chains of Length 7", "statement": "Are there any Shanks chains of length 7 with $p_{i+1} = 4p_i^2 - 17$?", "background": "Shanks chains are quadratic chains of primes. The recurrence $p_{i+1} = 4p_i^2 - 17$ yields a 4-chain if $p_1 = 3$ and a 5-chain if $p_1 = 303593$, but it can be seen (mod 59) that no such chain has length 17. It seems certain that such chains cannot be of arbitrary length. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A7.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set_id": 9 }, { "id": 1861, "problem_number": "GUY-A8a", "title": "Erdős $5000 Problem on Prime Gaps", "statement": "Is it true that for infinitely many $n$, $d_n = p_{n+1} - p_n > c \\ln n \\ln \\ln n \\ln \\ln \\ln \\ln n / (\\ln \\ln \\ln n)^2$ for arbitrarily large constant $c$?", "background": "Erdős offers $5,000 for a proof or disproof that the constant $c$ can be taken arbitrarily large. Rankin showed this holds for $c = e^\\gamma$, and Pintz improved it to $c = 2e^\\gamma > 3.562$. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A8.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set_id": 9 }, { "id": 1862, "problem_number": "GUY-A8b", "title": "Twin Prime Conjecture", "statement": "Are there infinitely many twin primes? That is, are there infinitely many primes $p$ such that $p + 2$ is also prime?", "background": "A very famous conjecture. Hardy and Littlewood conjectured that $P_2(n)$, the number of twin prime pairs less than $n$, is asymptotically $2cn/(\\ln n)^2$ where $2c \\approx 1.32032$. Brun showed that the sum of the reciprocals of twin primes is convergent. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A8.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set_id": 9 }, { "id": 1863, "problem_number": "GUY-A9", "title": "General Patterns of Consecutive Primes", "statement": "For any given pattern of primes with no congruence obstructions, are there infinitely many sets of consecutive primes with this pattern?", "background": "This conjecture is more general than Chowla's conjecture. It seems likely that there are infinitely many triples of primes $\\{6k - 1, 6k + 1, 6k + 5\\}$ and $\\{6k + 1, 6k + 5, 6k + 7\\}$. Hensley & Richards showed this is incompatible with the conjecture $\\pi(x + y) \\le \\pi(x) + \\pi(y)$ for all integers $x, y \\ge 2$. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A9.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set_id": 9 }, { "id": 1864, "problem_number": "GUY-A10", "title": "Gilbreath's Conjecture", "statement": "Define $d_n^k$ by $d_n^1 = p_{n+1} - p_n$ and $d_n^{k+1} = |d_{n+1}^k - d_n^k|$, the successive absolute differences of the sequence of primes. Is it true that $d_1^k = 1$ for all $k$?", "background": "Gilbreath conjectured this (and Proth claimed to have proved it long before). This was verified for $k < 63419$ by Killgrove & Ralston. Odlyzko checked it for primes up to $\\pi(10^{13})$. Croft and others suggest it has nothing to do with primes as such, but will be true for any sequence consisting of 2 and odd numbers which doesn't increase too fast. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A10.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set_id": 9 }, { "id": 1865, "problem_number": "GUY-A11", "title": "Erdős $100 Problem on Increasing and Decreasing Gaps", "statement": "Does there exist an $n_0$ such that for every $i$ and $n > n_0$ we have $d_{n+2i} > d_{n+2i+1}$ and $d_{n+2i+1} < d_{n+2i+2}$, where $d_n = p_{n+1} - p_n$?", "background": "Erdős & Turán showed that the values of $n$ for which $d_n > d_{n+1}$ have positive lower density, but it is not known if there are infinitely many increasing or decreasing sets of three consecutive values of $d_n$. Erdős offers $100.00 for a proof that such an $n_0$ does not exist. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A11.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set_id": 9 }, { "id": 1866, "problem_number": "GUY-A13", "title": "Erdős Conjecture on Carmichael Numbers", "statement": "Let $C(x)$ be the number of Carmichael numbers less than $x$. Does $(\\ln C(x))/\\ln x$ tend to 1 as $x$ tends to infinity?", "background": "Erdős conjectured this behavior for the count of Carmichael numbers. Alford, Granville & Pomerance showed there are infinitely many Carmichael numbers, in fact more than $x^\\beta$ of them less than $x$ for $\\beta > 0.290306$. Pomerance, Selfridge & Wagstaff give a heuristic argument supporting Erdős' conjecture. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A13.", "difficulty_level_id": 4, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set_id": 9 }, { "id": 1867, "problem_number": "GUY-A14a", "title": "Pomerance's Questions on Good Primes", "statement": "Call prime $p_n$ good if $p_n^2 > p_{n-i}p_{n+i}$ for all $i$, $1 \\le i \\le n-1$. Is it true that the set of $n$ for which $p_n$ is good has density 0? Are there infinitely many $n$ with $p_n p_{n+1} > p_{n-i} p_{n+1+i}$ for all $i$, $1 \\le i \\le n-1$?", "background": "Erdős and Straus introduced the concept of good primes. Examples include 5, 11, 17, and 29. Pomerance used the prime number graph to show there are infinitely many good primes and posed several related questions. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A14.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set_id": 9 }, { "id": 1868, "problem_number": "GUY-A15", "title": "Congruent Products of Consecutive Numbers", "statement": "What is the least prime $p$ such that there are integers $a, k_1, k_2, k_3$ with $\\prod_{i=1}^{k_1} (a+i) \\equiv \\prod_{i=1}^{k_2} (a+k_1+i) \\equiv \\prod_{i=1}^{k_3} (a+k_1+k_2+i) \\equiv 1 \\pmod{p}$?", "background": "Erdős observed that $3 \\cdot 4 \\equiv 5 \\cdot 6 \\cdot 7 \\equiv 1 \\pmod{11}$ and suggested that such primes $p$ exist for any number of congruent products. Narkiewicz and others found examples for larger numbers of terms. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A15.", "difficulty_level_id": 2, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set_id": 9 }, { "id": 1869, "problem_number": "GUY-A16", "title": "Walking to Infinity on Gaussian Primes", "statement": "Can one walk from the origin to infinity using Gaussian primes as stepping stones and taking steps of bounded length?", "background": "Motzkin and Gordon asked this question about Gaussian primes (primes in the ring of complex numbers $a+bi$ where $a, b$ are integers). Presumably not. Jordan & Rabung showed that steps of length at least 4 are necessary. Gethner, Wagon & Wick produced a moat of width $\\sqrt{26}$. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A16.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set_id": 9 }, { "id": 1870, "problem_number": "GUY-A17", "title": "Giuga's Conjecture on Prime Characterization", "statement": "Is it true that if $n$ divides $1^{n-1} + 2^{n-1} + \\dots + (n-1)^{n-1} + 1$, then $n$ is prime?", "background": "Sierpiński observed that if $n$ is prime, then $n$ divides this sum. Giuga conjectured the converse and verified it for $n \\le 10^{1000}$. A counterexample would be a Carmichael number with additional properties. An equivalent conjecture is $n B_{n-1} \\equiv -1 \\pmod{n}$ where $B_k$ are Bernoulli numbers. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A17.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set_id": 9 }, { "id": 1871, "problem_number": "GUY-A18", "title": "Erdős-Selfridge Classification: Infinitely Many Primes in Each Class", "statement": "In the Erdős-Selfridge classification of primes, are there infinitely many primes in each class? Prime $p$ is in class 1 if the only prime divisors of $p+1$ are 2 or 3; and $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\\le r-1$, with equality for at least one prime factor.", "background": "The first few classes are: Class 1: 2, 3, 5, 7, 11, 17, 23, 31, 47, 53, 71, 107, 127, 191, ...; Class 2: 13, 19, 29, 41, 43, 59, 61, 67, 79, 83, 89, 97, 101, ...; Class 3: 37, 103, 113, 151, 157, 163, 173, 181, 193, 227, 233, ... From Richard Guy's \"Unsolved Problems in Number Theory\", Section A18.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set_id": 9 }, { "id": 1872, "problem_number": "GUY-A19a", "title": "Erdős Conjecture on $n - 2^k$ Prime", "statement": "Are 4, 7, 15, 21, 45, 75, and 105 the only values of $n$ for which $n - 2^k$ is prime for all $k$ such that $2 \\le 2^k < n$?", "background": "Erdős conjectures that these are the only such values. He also conjectures that for infinitely many $n$, all the integers $n - 2^k, 1 \\le 2^k < n$ are squarefree. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A19.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set_id": 9 }, { "id": 1873, "problem_number": "GUY-A19b", "title": "Cohen-Selfridge Problem on $\\pm p^a \\pm 2^b$", "statement": "What is the least positive odd number not of the form $\\pm p^a \\pm 2^b$, where $p$ is an odd prime?", "background": "Cohen & Selfridge observed that the number is greater than $2^{18}$. This is related to the representation of odd numbers as sums or differences of prime powers and powers of 2. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A19.", "difficulty_level_id": 2, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set_id": 9 }, { "id": 1874, "problem_number": "GUY-A20", "title": "Density of Symmetric Primes", "statement": "Given pairs of odd primes $p, q$, define $S(q,p)$ as the number of lattice points $(m, n)$ in the rectangle $0 < m < p/2$, $0 < n < q/2$ below the diagonal. A pair is symmetric if $S(p, q) = S(q,p)$. Is the number of symmetric primes less than $x$ equal to $x/(\\ln x)^{\\sigma+o(1)}$, where $\\sigma = 2 - (1+\\ln \\ln 2)/\\ln 2 \\approx 1.08607$?", "background": "Fletcher, Lindgren & Pomerance showed that a pair is symmetric just if $|p - q| = (p - 1, q - 1)$, and that the number of symmetric primes less than $x$ is at most $x/(\\ln x)^{1.027}$. They conjectured the more precise asymptotic. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A20.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set_id": 9 }, { "id": 1875, "problem_number": "GUY-A12a", "title": "Square Pseudoprimes", "statement": "Are there any square pseudoprimes (base 2) other than multiples of $1194649 = 1093^2$ or $12327121 = 3511^2$?", "background": "Pinch observed that there are 54 non-squarefree pseudoprimes up to $10^{13}$, all multiples of $1093^2$ or $3511^2$. The question asks if there are other perfect squares that are pseudoprimes to base 2. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A12.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set_id": 9 }, { "id": 1876, "problem_number": "GUY-A12b", "title": "Selfridge-Wagstaff-Pomerance Prize Problem", "statement": "Does there exist a composite number $n \\equiv 3$ or $7 \\pmod{10}$ which divides both $2^n - 2$ and the Fibonacci number $u_{n+1}$?", "background": "Selfridge, Wagstaff & Pomerance offer $500 + $100 + $20 = $620 for finding such a composite $n$, or $20 + $100 + $500 = $620 for a proof that no such $n$ exists. This combines pseudoprime properties with Fibonacci divisibility. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A12.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set_id": 9 }, { "id": 1877, "problem_number": "GUY-A12c", "title": "Even Fibonacci Pseudoprimes", "statement": "Does there exist an even Fibonacci pseudoprime?", "background": "A Fibonacci pseudoprime of the $m$-th kind is an odd composite integer $n$ with $V_n(m, -1) \\equiv m \\pmod n$ where $V_n$ is the Lucas sequence. Somer showed that if an even Fibonacci pseudoprime exists, it must be greater than $28 \\times 10^{12}$. From Richard Guy's \"Unsolved Problems in Number Theory\", Section A12.", "difficulty_level_id": 3, "status": "open", "category_id": 1, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set_id": 9 }, { "id": 1878, "problem_number": "EP-1", "title": "Erdős Problem #1", "statement": "If $A\\subseteq \\{1,\\ldots,N\\}$ with $\\lvert A\\rvert=n$ is such that the subset sums $\\sum_{a\\in S}a$ are distinct for all $S\\subseteq A$ then $ N \\gg 2^{n}. $ ", "background": "Erdos called this 'perhaps my first serious problem' (in \\cite{Er98} he dates it to 1931). The powers of $2$ show that $2^n$ would be best possible here. The trivial lower bound is $N \\gg 2^{n}/n$, since all $2^n$ distinct subset sums must lie in $[0,Nn)$. Erdos and Moser \\cite{Er56} proved $ N\\geq (\\tfrac{1}{4}-o(1))\\frac{2^n}{\\sqrt{n}}. $ (In \\cite{Er85c} Erdos offered \\$100 for any improvement of the constant $1/4$ here.)\nA number of improvements of the constant have been given (see \\cite{St23} for a history), with the current record $\\sqrt{2/\\pi}$ first proved in unpublished work of Elkies and Gleason. Two proofs achieving this constant are provided by Dubroff, Fox, and Xu \\cite{DFX21}, who in fact prove the exact bound $N\\geq \\binom{n}{\\lfloor n/2\\rfloor}$.\nIn \\cite{Er73} and \\cite{ErGr80} the generalisation where $A\\subseteq (0,N]$ is a set of real numbers such that the subset sums all differ by at least $1$ is proposed, with the same conjectured bound. (The second proof of \\cite{DFX21} applies also to this generalisation.) This generalisation seems to have first appeared in \\cite{Gr71}.\nThis problem appears in Erdos' book with Spencer \\cite{ErSp74} in the final chapter titled 'The kitchen sink'. As Ruzsa writes in \\cite{Ru99} \"it is a rich kitchen where such things go to the sink\".\nThe sequence of minimal $N$ for a given $n$ is A276661 in the OEIS.\nSee also [350].\nThis is discussed in problem C8 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[DFX21] Dubroff, Q. and Fox, J. and Xu, M. W., A note on the Erdos distinct subset sums problem. SIAM Journal on Discrete Mathematics (2021), 322-324.\n\n[Er56] Erdos, P., Problems and results in additive number theory. Colloque sur la Th\\'{e}orie des Nombres, Bruxelles, 1955 (1956), 127-137.\n\n[Er73] Erdos, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138.\n\n[Er85c] Erdos, P., On some of my problems in number theory I would most like to see solved. Number theory (Ootacamund, 1984) (1985), 74-84.\n\n[Er98] Erdos, Paul, Some of my new and almost new problems and results in combinatorial number theory. Number theory (Eger, 1996) (1998), 169-180.\n\n[ErGr80] Erdos, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).\n\n[ErSp74] Erdos, Paul and Spencer, Joel, Probabilistic methods in combinatorics. Akad\\'{e}miai Kiad\\'{o} (1974).\n\n[Gr71] Graham, R. L., On sums of integers taken from a fixed sequence. (1971), 22--40.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Ru99] Ruzsa, I., Erdos and the Integers. Journal of Number Theory (1999), 115-163.\n\n[St23] Steinerberger, S., Some remarks on the Erdos distinct subset sums problem. arXiv:2208.12182 (2023).\",\n \"difficulty\": \"L3\"\n},\n{", "difficulty_level_id": 3, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1879, "problem_number": "EP-3", "title": "Erdős Problem #3", "statement": "If $A\\subseteq \\mathbb{N}$ has $\\sum_{n\\in A}\\frac{1}{n}=\\infty$ then must $A$ contain arbitrarily long arithmetic progressions?", "background": "This is essentially asking for good bounds on $r_k(N)$, the size of the largest subset of $\\{1,\\ldots,N\\}$ without a non-trivial $k$-term arithmetic progression. For example, a bound like $ r_k(N) \\ll_k \\frac{N}{(\\log N)(\\log\\log N)^2} $ would be sufficient.\nEven the case $k=3$ is non-trivial, but was proved by Bloom and Sisask \\cite{BlSi20}. Much better bounds for $r_3(N)$ were subsequently proved by Kelley and Meka \\cite{KeMe23}. Green and Tao \\cite{GrTa17} proved $r_4(N)\\ll N/(\\log N)^{c}$ for some small constant $c>0$. Gowers \\cite{Go01} proved $ r_k(N) \\ll \\frac{N}{(\\log\\log N)^{c_k}}, $ where $c_k>0$ is a small constant depending on $k$. The current best bounds for general $k$ are due to Leng, Sah, and Sawhney \\cite{LSS24}, who show that $ r_k(N) \\ll \\frac{N}{\\exp((\\log\\log N)^{c_k})} $ for some constant $c_k>0$ depending on $k$.\nCuriously, Erdos \\cite{Er83c} thought this conjecture was the 'only way to approach' the conjecture that there are arbitrarily long arithmetic progressions of prime numbers, now a theorem due to Green and Tao \\cite{GrTa08} (see [219]).\nIn \\cite{Er81} Erdos makes the stronger conjecture that $ r_k(N) \\ll_C\\frac{N}{(\\log N)^C} $ for every $C>0$ (now known for $k=3$ due to Kelley and Meka \\cite{KeMe23}) - see [140].\nSee also [139] and [142].\nThis is discussed in problem A5 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[BlSi20] Bloom, T.F. and Sisask, O., Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions. arXiv:2007.03528 (2020).\n\n[Er81] Erdos, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.\n\n[Er83c] Erdos, Paul, Combinatorial problems in geometry. Math. Chronicle (1983), 35-54.\n\n[Go01] Gowers, W. T., A new proof of Szemer\\'{e}di's theorem. Geom. Funct. Anal. (2001), 465-588.\n\n[GrTa08] Green, Ben and Tao, Terence, The primes contain arbitrarily long arithmetic progressions. Ann. of Math. (2) (2008), 481-547.\n\n[GrTa17] Green, Ben and Tao, Terence, New bounds for Szemer\\'{e}di's theorem, III: a polylogarithmic bound for $r_4(N)$. Mathematika (2017), 944-1040.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[KeMe23] Kelley, Z. and Meka, R., Strong Bounds for 3-Progressions. arXiv:2302.05537 (2023).\n\n[LSS24] Leng, J., Sah, A. and Sawhney, M., Improved bounds for Szemer\\'{e}di's theorem. arXiv:2402.17995 (2024).", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1880, "problem_number": "EP-5", "title": "Erdős Problem #5", "statement": "Let $C\\geq 0$. Is there an infinite sequence of $n_i$ such that $ \\lim_{i\\to \\infty}\\frac{p_{n_i+1}-p_{n_i}}{\\log n_i}=C? $ ", "background": "Let $S$ be the set of limit points of $(p_{n+1}-p_n)/\\log n$. This problem asks whether $S=[0,\\infty]$. Although this conjecture remains unproven, a lot is known about $S$. Some highlights:\n{UL}\n{LI}$\\infty\\in S$ by Westzynthius' result \\cite{We31} on large prime gaps,{/LI}\n{LI}$0\\in S$ by the work of Goldston, Pintz, and Yildirim \\cite{GPY09} on small prime gaps,{/LI}\n{LI}Erdos \\cite{Er55} and Ricci \\cite{Ri56} independently showed that $S$ has positive Lebesgue measure,{/LI}\n{LI} Hildebrand and Maier \\cite{HiMa88} showed that $S$ contains arbitrarily large (finite) numbers,{/LI}\n{LI} Pintz \\cite{Pi16} showed that there exists some small constant $c>0$ such that $[0,c]\\subset S$,{/LI}\n{LI} Banks, Freiberg, and Maynard \\cite{BFM16} showed that at least $12.5\\%$ of $[0,\\infty)$ belongs to $S$,{/LI}\n{LI} Merikoski \\cite{Me20} showed that at least $1/3$ of $[0,\\infty)$ belongs to $S$, and that $S$ has bounded gaps.{/LI}\n{/UL}\nIn \\cite{Er65b}, \\cite{Er85c}, and \\cite{Er97c} Erdos asks whether $S$ is everywhere dense (but Weisenberg notes that clearly $S$ is closed so this is equivalent to asking whether $S=[0,\\infty]$).\nSee also [234].\nReferences\n\n\n[BFM16] Banks, William D. and Freiberg, Tristan and Maynard, James, On limit points of the sequence of normalized prime gaps. Proc. Lond. Math. Soc. (3) (2016), 515-539.\n\n[Er55] Erd\"{o}s, Paul, Some remarks on number theory. Riveon Lematematika (1955), 45-48.\n\n[Er65b] Erdos, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244.\n\n[Er85c] Erdos, P., On some of my problems in number theory I would most like to see solved. Number theory (Ootacamund, 1984) (1985), 74-84.\n\n[Er97c] Erdos, Paul, Some of my favorite problems and results. The mathematics of Paul Erdos, I (1997), 47-67.\n\n[GPY09] Goldston, Daniel A. and Pintz, J\\'{a}nos and Y\\i ld\\i r\\i m, Cem Y., Primes in tuples. I. Ann. of Math. (2) (2009), 819-862.\n\n[HiMa88] Hildebrand, Adolf and Maier, Helmut, Gaps between prime numbers. Proc. Amer. Math. Soc. (1988), 1-9.\n\n[Me20] Merikoski, Jori, Limit points of normalized prime gaps. J. Lond. Math. Soc. (2) (2020), 99-124.\n\n[Pi16] Pintz, J\\'{a}nos, Polignac numbers, conjectures of Erdos on gaps between primes, arithmetic progressions in primes, and the bounded gap conjecture. From arithmetic to zeta-functions (2016), 367-384.\n\n[Ri56] Ricci, Giovanni, Recherches sur l'allure de la suite $\\{p_{n+1}-p_n/\\log p_n\\}$. Colloque sur la Th\\'{e}orie des Nombres, Bruxelles, 1955 (1956), 93-106.\n\n[We31] Westzynthius, E., \"{U}ber die Verteilung der Zahlen, die zu den n ersten Primzahlen teilerfremd sind. Commentat. Phys. Math. (1931), 1-37.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1881, "problem_number": "EP-9", "title": "Erdős Problem #9", "statement": "Let $A$ be the set of all odd integers not of the form $p+2^{k}+2^l$ (where $k,l\\geq 0$ and $p$ is prime). Is the upper density of $A$ positive?", "background": "In \\cite{Er77c} Erdos credits Schinzel with proving that there are infinitely many odd integers not of this form, but gives no reference. Crocker \\cite{Cr71} has proved there are $\\gg\\log\\log N$ such integers in $\\{1,\\ldots,N\\}$. Pan \\cite{Pa11} improved this to $\\gg_\\epsilon N^{1-\\epsilon}$ for any $\\epsilon>0$. Erdos believed this cannot be proved by covering systems, i.e. integers of the form $p+2^k+2^l$ exist in every infinite arithmetic progression.\nThe sequence of such numbers is A006286 in the OEIS.\nSee also [10], [11], and [16].\nThis is discussed in problem A19 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Cr71] Crocker, Roger, On the sum of a prime and of two powers of two. Pacific J. Math. (1971), 103-107.\n\n[Er77c] Erdos, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ.,\nNew York, 1976) (1977), 43-72.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Pa11] Pan, Hao, On the integers not of the form {$p+2^a+2^b$}. Acta Arith. (2011), 55-61.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1882, "problem_number": "EP-10", "title": "Erdős Problem #10", "statement": "Is there some $k$ such that every integer is the sum of a prime and at most $k$ powers of 2?", "background": "Erdos described this as 'probably unattackable'. In \\cite{ErGr80} Erdos and Graham suggest that no such $k$ exists. Gallagher \\cite{Ga75} has shown that for any $\\epsilon>0$ there exists $k(\\epsilon)$ such that the set of integers which are the sum of a prime and at most $k(\\epsilon)$ many powers of 2 has lower density at least $1-\\epsilon$.\nGranville and Soundararajan \\cite{GrSo98} have conjectured that at most $3$ powers of 2 suffice for all odd integers, and hence at most $4$ powers of $2$ suffice for all even integers. (The restriction to odd integers is important here - for example, Bogdan Grechuk has observed that $1117175146$ is not the sum of a prime and at most $3$ powers of $2$, and pointed out that parity considerations, coupled with the fact that there are many integers not the sum of a prime and $2$ powers of $2$ (see [9]) suggest that there exist infinitely many even integers which are not the sum of a prime and at most $3$ powers of $2$).\nSee also [9], [11], and [16].\nThis problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.\nReferences\n\n\n[ErGr80] Erdos, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).\n\n[Ga75] Gallagher, P. X., Primes and powers of 2. Invent. Math. (1975), 125-142.\n\n[GrSo98] Granville, A. and Soundararajan, K., A Binary Additive Problem of Erdos and the Order of $2$ mod $p^2$. The Ramanujan Journal (1998), 283-298.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1883, "problem_number": "EP-12", "title": "Erdős Problem #12", "statement": "Let $A$ be an infinite set such that there are no distinct $a,b,c\\in A$ such that $a\\mid (b+c)$ and $b,c>a$. Is there such an $A$ with $ \\liminf \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N^{1/2}}>0? $ Does there exist some absolute constant $c>0$ such that there are always infinitely many $N$ with $ \\lvert A\\cap\\{1,\\ldots,N\\}\\rvert\\frac{N}{f(N)}. $ (Their example is given by all integers in $(y_i,\\frac{3}{2}y_i)$ congruent to $1$ modulo $(2y_{i-1})!$, where $y_i$ is some sufficiently quickly growing sequence.)\nAn example of an $A$ with this property where $ \\liminf \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N^{1/2}}\\log N>0 $ is given by the set of $p^2$, where $p\\equiv 3\\pmod{4}$ is prime.\nElsholtz and Planitzer \\cite{ElPl17} have constructed such an $A$ with $ \\lvert A\\cap\\{1,\\ldots,N\\}\\rvert\\gg \\frac{N^{1/2}}{(\\log N)^{1/2}(\\log\\log N)^2(\\log\\log\\log N)^2}. $ Schoen \\cite{Sc01} proved that if all elements in $A$ are pairwise coprime then $ \\lvert A\\cap\\{1,\\ldots,N\\}\\rvert \\ll N^{2/3} $ for infinitely many $N$. Baier \\cite{Ba04} has improved this to $\\ll N^{2/3}/\\log N$.\nFor the finite version see [13].\nThis problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.\nReferences\n\n\n[Ba04] Baier, Stephan, A note on {$\\scr P$}-sets. Integers (2004), A13, 6.\n\n[ElPl17] Elsholtz, Christian and Planitzer, Stefan, On Erdos and {S}\\'{a}rk\"ozy's sequences with Property P. Monatsh. Math. (2017), 565--575.\n\n[ErSa70] Erdos, P. and S\\'{a}rk\"ozi, A., On the divisibility properties of sequences of integers. Proc. London Math. Soc. (3) (1970), 97-101.\n\n[Sc01] Schoen, Tomasz, On a problem of Erdos and {S}\\'{a}rk\"ozy. J. Combin. Theory Ser. A (2001), 191--195.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1884, "problem_number": "EP-14", "title": "Erdős Problem #14", "statement": "Let $A\\subseteq \\mathbb{N}$. Let $B\\subseteq \\mathbb{N}$ be the set of integers which are representable in exactly one way as the sum of two elements from $A$.\nIs it true that for all $\\epsilon>0$ and large $N$ $ \\lvert \\{1,\\ldots,N\\}\\backslash B\\rvert \\gg_\\epsilon N^{1/2-\\epsilon}? $ Is it possible that $ \\lvert \\{1,\\ldots,N\\}\\backslash B\\rvert =o(N^{1/2})? $ ", "background": "Apparently originally considered by Erdos and Nathanson, although later Erdos attributes this to Erdos, S\\'{a}rk\"{o}zy, and Szemer\\'{e}di (but gives no reference), and claims a construction of an $A$ such that for all $\\epsilon>0$ and all large $N$ $ \\lvert \\{1,\\ldots,N\\}\\backslash B\\rvert \\ll_\\epsilon N^{1/2+\\epsilon}, $ and yet there for all $\\epsilon>0$ there exist infinitely many $N$ where $ \\lvert \\{1,\\ldots,N\\}\\backslash B\\rvert \\gg_\\epsilon N^{1/3-\\epsilon}. $ Erd\"{o}s and Freud investigated the finite analogue in \\cite{ErFr91}, proving that there exists $A\\subseteq \\{1,\\ldots,N\\}$ such that the number of integers not representable in exactly one way as the sum of two elements from $A$ is $<2^{3/2}N^{1/2}$, and suggest the constant $2^{3/2}$ is perhaps best possible.\nReferences\n\n\n[ErFr91] Erdos, P. and Freud, R., On sums of a {S}idon-sequence. J. Number Theory (1991), 196--205.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1885, "problem_number": "EP-15", "title": "Erdős Problem #15", "statement": "Is it true that $ \\sum_{n=1}^\\infty(-1)^n\\frac{n}{p_n} $ converges, where $p_n$ is the sequence of primes?", "background": "Erdos suggested that a computer could be used to explore this, and did not see any other method to attack this.\nTao \\cite{Ta23} has proved that this series does converge assuming a strong form of the Hardy-Littlewood prime tuples conjecture.\nIn \\cite{Er98} Erdos further conjectures that $ \\sum_{n=1}^\\infty (-1)^n \\frac{1}{n(p_{n+1}-p_n)} $ converges and $ \\sum_{n=1}^\\infty (-1)^n \\frac{1}{p_{n+1}-p_n} $ diverges. Weisenberg notes that the existence of infinitely many bounded gaps between primes (as proved by Zhang \\cite{Zh14}) implies the latter series does not converge. Weisenberg also has an argument which shows that, assuming the Hardy-Littlewood prime $k$-tuples conjecture, the series is unbounded in at least one direction (positive or negative).\nErdos further conjectured that $ \\sum_{n=1}^\\infty (-1)^n \\frac{1}{n(p_{n+1}-p_n)(\\log\\log n)^c} $ converges for every $c>0$, and reports that he and Nathanson can prove that this series converges absolutely for $c>2$ (and can show, conditional on 'hopeless' conjectures about the primes, that this sum does not converge absolutely for $c=2$).\nSawhney has provided the following proof that this series converges absolutely for $c>2$: note that, whenever $c>1$, the contribution to the sum from gaps $p_{n+1}-p_n\\geq \\log n$ is convergent, so it suffices to consider only small gaps. The number of $n\\leq X$ such that $p_{n+1}-p_n\\in [\\epsilon\\log n,2\\epsilon \\log n)$ is bounded above by $\\ll \\epsilon X$ (this can be proved via the Selberg sieve). In particular, applying this bound for $\\frac{1}{\\log n}\\leq \\epsilon \\leq 1$ of the shape $2^{-j}$ (of which there are at most $\\log\\log n$ possibilities) shows the desired convergence, since $ \\sum \\frac{1}{n(\\log n)(\\log\\log n)^{c-1}} $ converges.\nReferences\n\n\n[Er98] Erdos, Paul, Some of my new and almost new problems and results in combinatorial number theory. Number theory (Eger, 1996) (1998), 169-180.\n\n[Ta23] Tao, T., The convergence of an alternating series of Erdos, assuming the Hardy-Littlewood prime tuples conjecture. arXiv:2308.07205 (2023).\n\n[Zh14] Zhang, Yitang, Bounded gaps between primes. Ann. of Math. (2) (2014), 1121--1174.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1886, "problem_number": "EP-17", "title": "Erdős Problem #17", "statement": "Are there infinitely many primes $p$ such that every even number $n\\leq p-3$ can be written as a difference of primes $n=q_1-q_2$ where $q_1,q_2\\leq p$?", "background": "The first prime without this property is $97$. The sequence of such primes is A038133 in the OEIS. These are called cluster primes.\nBlecksmith, Erdos, and Selfridge \\cite{BES99} proved that the number of such primes is $ \\ll_A \\frac{x}{(\\log x)^A} $ for every $A>0$, and Elsholtz \\cite{El03} improved this to $ \\ll x\\exp(-c(\\log\\log x)^2) $ for every $c<1/8$.\nThis is discussed in problem C1 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[BES99] Blecksmith, Richard and Erdos, Paul and Selfridge, J. L., Cluster primes. Amer. Math. Monthly (1999), 43--48.\n\n[El03] Elsholtz, Christian, On cluster primes. Acta Arith. (2003), 281--284.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1887, "problem_number": "EP-18", "title": "Erdős Problem #18", "statement": "We call $m$ practical if every integer $n0$?", "background": "Erdos and Rado \\cite{ErRa60} originally proved $f(n,k)\\leq (k-1)^nn!$. Kostochka \\cite{Ko97} improved this slightly (in particular establishing an upper bound of $o(n!)$, for which Erdos awarded him the consolation prize of \\$100), but the bound stood at $n^{(1+o(1))n}$ for a long time until Alweiss, Lovett, Wu, and Zhang \\cite{ALWZ20} proved $ f(n,k) < (Ck\\log n\\log\\log n)^n $ for some constant $C>1$. This was refined slightly, independently by Rao \\cite{Ra20}, Frankston, Kahn, Narayanan, and Park \\cite{FKNP19}, and Bell, Chueluecha, and Warnke \\cite{BCW21}, leading to the current record of $ f(n,k) < (Ck\\log n)^n $ for some constant $C>1$.\nIn \\cite{Er81} offered \\$1000 for a proof or disproof even just in the special case when $k=3$, which he expected 'contains the whole difficulty'. He also wrote 'I really do not see why this question is so difficult'.\nThe usual focus is on the regime where $k=O(1)$ is fixed (say $k=3$) and $n$ is large, although for the opposite regime Kostochka, R\"{o}dl, and Talysheva \\cite{KRT99} have shown $ f(n,k)=(1+O_n(k^{-1/2^n}))k^n. $ \nReferences\n\n\n[ALWZ20] Alweiss, R. and Lovett, S. and Wu, K. and Zhang, J., Improved bounds for the sunflower lemma. (2020).\n\n[BCW21] Bell, T. and Chueluecha, S. and Warnke, L., Note on sunflowers. Discret. Math. (2021).\n\n[Er81] Erdos, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.\n\n[ErRa60] Erdos, P. and Rado, R., Intersection theorems for systems of sets. J. London Math. Soc. (1960), 85-90.\n\n[FKNP19] Frankston, K. and Kahn, J. and Narayanan, B. and Park, J., Thresholds versus fractional expectation-thresholds. CoRR (2019).\n\n[KRT99] Kostochka, A. V. and R\"{o}dl, V. and Talysheva, L. A., On systems of small sets with no large $\\Delta$-subsystems. Combin. Probab. Comput. (1999), 265-268.\n\n[Ko97] Kostochka, A., A bound on the cardinality of families not containing $\\Delta$-systems. (1997).\n\n[Ra20] Rao, A., Coding for sunflowers. Discrete Analysis (2020).", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1889, "problem_number": "EP-25", "title": "Erdős Problem #25", "statement": "Let $n_10$.\nAnother stronger conjecture would be that the hypothesis $\\lvert A\\cap [1,N]\\rvert \\gg N^{1/2}$ for all large $N$ suffices.\nErdos and S\\'{a}rk\"{o}zy conjectured the stronger version that if $A=\\{a_10$, $ h(N) = N^{1/2}+O_\\epsilon(N^\\epsilon)? $ ", "background": "A problem of Erdos and Tur\\'{a}n. It may even be true that $h(N)=N^{1/2}+O(1)$, but Erdos remarks this is perhaps too optimistic. Erdos and Tur\\'{a}n \\cite{ErTu41} proved an upper bound of $N^{1/2}+O(N^{1/4})$, with an alternative proof by Lindstr\"{o}m \\cite{Li69}. Both proofs in fact give $ h(N) \\leq N^{1/2}+N^{1/4}+1. $ Balogh, F\"{u}redi, and Roy \\cite{BFR21} improved the bound in the error term to $0.998N^{1/4}$. This was further optimised by O'Bryant \\cite{OB22}. The current record is $ h(N)\\leq N^{1/2}+0.98183N^{1/4}+O(1), $ due to Carter, Hunter, and O'Bryant \\cite{CHO25}.\nSinger \\cite{Si38} was the first to show that $h(N)\\geq (1-o(1))N^{1/2}$ for all $N$. For a detailed survey of the literature we refer to \\cite{OB04}.\nSee also [241] and [840].\nThis problem is Problem 31 on Green's open problems list.\nThis is discussed in problem C9 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[BFR21] Balogh, J. and F\"{u}redi, Z. and Roy, S., An upper bound on the size of Sidon sets. arXiv:2103.15850 (2021).\n\n[CHO25] Carter, D. and Hunter, Z. and O'Bryant, K., On the diameter of finite {S}idon sets. Acta Math. Hungar. (2025), 108--126.\n\n[ErTu41] Erdos, P. and Tur\\'{a}n, P., On a problem of Sidon in additive number theory, and on some related problems. J. London Math. Soc. (1941), 212-215.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Li69] Lindstr\"{o}m, B., An inequality for $B_2$-sequences. J. Combinatorial Theory (1969), 211-212.\n\n[OB04] O'Bryant, Kevin, A complete annotated bibliography of work related to {S}idon\nsequences. Electron. J. Combin. (2004), 39.\n\n[OB22] O'Bryant, K., On the size of finite Sidon sets. arXiv:2207.07800 (2022).\n\n[Si38] Singer, James, A theorem in finite projective geometry and some applications\nto number theory. Trans. Amer. Math. Soc. (1938), 377--385.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1892, "problem_number": "EP-32", "title": "Erdős Problem #32", "statement": "Is there a set $A\\subset\\mathbb{N}$ such that $ \\lvert A\\cap\\{1,\\ldots,N\\}\\rvert = o((\\log N)^2) $ and such that every large integer can be written as $p+a$ for some prime $p$ and $a\\in A$?\nCan the bound $O(\\log N)$ be achieved? Must such an $A$ satisfy $ \\liminf \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{\\log N}> 1? $ ", "background": "Such a set is called an additive complement to the primes.\nErdos \\cite{Er54} proved that such a set $A$ exists with $\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert\\ll (\\log N)^2$ (improving a previous result of Lorentz \\cite{Lo54} who achieved $\\ll (\\log N)^3$).\nWolke \\cite{Wo96} has shown that such a bound is almost true, in that we can achieve $\\ll (\\log N)^{1+o(1)}$ if we only ask for almost all integers to be representable. Kolountzakis \\cite{Ko96} improved this to $\\ll (\\log N)(\\log\\log N)$, and Ruzsa \\cite{Ru98c} further improved this to $\\ll \\omega(N)\\log N$ for any $\\omega\\to \\infty$.\nThe answer to the third question is yes: Ruzsa \\cite{Ru98c} has shown that we must have $ \\liminf \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{\\log N}\\geq e^\\gamma\\approx 1.781. $ This is discussed in problem E1 of Guy's collection \\cite{Gu04}, where it is stated that Erdos offered \\$50 for determining whether $O(\\log N)$ can be achieved.\nReferences\n\n\n[Er54] Erdos, Paul, Some results on additive number theory. Proc. Amer. Math. Soc. (1954), 847-853.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Ko96] Kolountzakis, Mihail N., On the additive complements of the primes and sets of similar\ngrowth. Acta Arith. (1996), 1--8.\n\n[Lo54] Lorentz, G. G., On a problem of additive number theory. Proc. Amer. Math. Soc. (1954), 838-841.\n\n[Ru98c] Ruzsa, Imre Z., On the additive completion of primes. Acta Arith. (1998), 269-275.\n\n[Wo96] Wolke, Dieter, On a problem of Erdos in additive number theory. J. Number Theory (1996), 209-213.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1893, "problem_number": "EP-33", "title": "Erdős Problem #33", "statement": "Let $A\\subset\\mathbb{N}$ be such that every large integer can be written as $n^2+a$ for some $a\\in A$ and $n\\geq 0$. What is the smallest possible value of $ \\limsup \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N^{1/2}}? $ Is $ \\liminf \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N^{1/2}}>1? $ ", "background": "Such a set $A$ is called an additive complement of the set of squares. Erdos observed that there exist $A$ for which the $\\limsup$ is finite and $>1$. Moser \\cite{Mo65} proved that, for any such $A$, $ \\liminf \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N^{1/2}}>1.06. $ The best-known lower bound is $ \\liminf \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N^{1/2}}\\geq\\frac{4}{\\pi}\\approx 1.273 $ proved by Cilleruelo \\cite{Ci93}, Habsieger \\cite{Ha95}, and Balasubramanian and Ramana \\cite{BaRa01}.\nThe problem of minimising the $\\limsup$ appears to have been much less studied. van Doorn has a construction of such an $A$ in which, for all $N$, $ \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N^{1/2}}< 2\\phi^{5/2}\\approx 6.66, $ where $\\phi=\\frac{1+\\sqrt{5}}{2}$ is the golden ratio.\nReferences\n\n\n[BaRa01] Balasubramanian, R. and Ramana, D. S., Additive complements of the squares. C. R. Math. Acad. Sci. Soc. R. Can. (2001), 6--11.\n\n[Ci93] Cilleruelo, Javier, The additive completion of {$k$}th-powers. J. Number Theory (1993), 237--243.\n\n[Ha95] Habsieger, Laurent, On the additive completion of polynomial sets. J. Number Theory (1995), 130--135.\n\n[Mo65] Moser, Leo, On the additive completion of sets of integers. (1965), 175--180.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1894, "problem_number": "EP-36", "title": "Erdős Problem #36", "statement": "Find the optimal constant $c>0$ such that the following holds.\nFor all sufficiently large $N$, if $A\\sqcup B=\\{1,\\ldots,2N\\}$ is a partition into two equal parts, so that $\\lvert A\\rvert=\\lvert B\\rvert=N$, then there is some $x$ such that the number of solutions to $a-b=x$ with $a\\in A$ and $b\\in B$ is at least $cN$.", "background": "The minimum overlap problem. The example (with $N$ even) $A=\\{N/2+1,\\ldots,3N/2\\}$ shows that $c\\leq 1/2$ (indeed, Erdos initially conjectured that $c=1/2$). The lower bound of $c\\geq 1/4$ is trivial, and Scherk improved this to $1-1/\\sqrt{2}=0.29\\cdots$. The current records are $ 0.379005 < c < 0.380924, $ the lower bound due to White \\cite{Wh22} and the upper bound due to AlphaEvolve \\cite{GGTW25}, improving slightly on an upper bound due to Haugland \\cite{Ha16}.\nThis is discussed in problem C17 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[GGTW25] B. Georgiev, J. G\\'{o}mez-Serrano, T. Tao, and A. Wagner, Mathematical exploration and discovery at scale. arXiv:2511.02864 (2025).\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Ha16] Haugland, J. K., The minimum overlap problem revisited. arXiv:1609.08000 (2016).\n\n[Wh22] White, E. P., Erdos' minimum overlap problem. arXiv:2201.05704 (2022).", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1895, "problem_number": "EP-38", "title": "Erdős Problem #38", "statement": "Does there exist $B\\subset\\mathbb{N}$ which is not an additive basis, but is such that for every set $A\\subseteq\\mathbb{N}$ of Schnirelmann density $\\alpha$ and every $N$ there exists $b\\in B$ such that $ \\lvert (A\\cup (A+b))\\cap \\{1,\\ldots,N\\}\\rvert\\geq (\\alpha+f(\\alpha))N $ where $f(\\alpha)>0$ for $0<\\alpha <1 $?\nThe Schnirelmann density is defined by $ d_s(A) = \\inf_{N\\geq 1}\\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N}. $ ", "background": "Erdos \\cite{Er36c} proved that if $B$ is an additive basis of order $k$ then, for any set $A$ of Schnirelmann density $\\alpha$, for every $N$ there exists some integer $b\\in B$ such that $ \\lvert (A\\cup (A+b))\\cap \\{1,\\ldots,N\\}\\rvert\\geq \\left(\\alpha+\\frac{\\alpha(1-\\alpha)}{2k}\\right)N. $ It seems an interesting question (not one that Erdos appears to have asked directly, although see [35]) to improve the lower bound here, even in the case $B=\\mathbb{N}$. Erdos observed that a random set of density $\\alpha$ shows that the factor of $\\frac{\\alpha(1-\\alpha)}{2}$ in this case cannot be improved past $\\alpha(1-\\alpha)$.\nThis is a stronger property than $B$ being an essential component (see [37]). Linnik \\cite{Li42} gave the first construction of an essential component which is not an additive basis.\nReferences\n\n\n[Er36c] Erdos, P., On the arithmetical density of the sum of two sequences, one of which forms a basis for the integers. Acta. Arith. (1936), 201-207.\n\n[Li42] Linnik, U. V., On Erd\"{o}s's theorem on the addition of numerical sequences. Rec. Math. [Mat. Sbornik] N.S. (1942), 67-78.", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1896, "problem_number": "EP-39", "title": "Erdős Problem #39", "statement": "Is there an infinite Sidon set $A\\subset \\mathbb{N}$ such that $ \\lvert A\\cap \\{1\\ldots,N\\}\\rvert \\gg_\\epsilon N^{1/2-\\epsilon} $ for all $\\epsilon>0$?", "background": "The trivial greedy construction achieves $\\gg N^{1/3}$. The first improvement on this was achieved by Ajtai, Koml\\'{o}s, and Szemer\\'{e}di \\cite{AKS81b}, who found an infinite Sidon set with growth rate $\\gg (N\\log N)^{1/3}$. The current best bound of $\\gg N^{\\sqrt{2}-1+o(1)}$ is due to Ruzsa \\cite{Ru98}.\nErdos \\cite{Er73} had offered \\$25 for any construction which achieves $N^{c}$ for some $c>1/3$. Later he \\cite{Er77c} offered \\$100 for a construction which achieves $\\omega(N)N^{1/3}$ for some $\\omega(N)\\to \\infty$.\nErdos proved that for every infinite Sidon set $A$ we have $ \\liminf \\frac{\\lvert A\\cap \\{1,\\ldots,N\\}\\rvert}{N^{1/2}}=0. $ Erdos and R\\'{e}nyi have constructed, for any $\\epsilon>0$, a set $A$ such that $ \\lvert A\\cap \\{1\\ldots,N\\}\\rvert \\gg_\\epsilon N^{1/2-\\epsilon} $ for all large $N$ and $1_A\\ast 1_A(n)\\ll_\\epsilon 1$ for all $n$.\nThis is discussed in problem C9 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[AKS81b] Ajtai, Mikl\\'os and Koml\\'os, J\\'anos and Szemer\\'{e}di, Endre, A dense infinite {S}idon sequence. European J. Combin. (1981), 1--11.\n\n[Er73] Erdos, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138.\n\n[Er77c] Erdos, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ.,\nNew York, 1976) (1977), 43-72.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Ru98] Ruzsa, Imre Z., An infinite Sidon sequence. J. Number Theory (1998), 63-71.", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1897, "problem_number": "EP-40", "title": "Erdős Problem #40", "statement": "For what functions $g(N)\\to \\infty$ is it true that $ \\lvert A\\cap \\{1,\\ldots,N\\}\\rvert \\gg \\frac{N^{1/2}}{g(N)} $ implies $\\limsup 1_A\\ast 1_A(n)=\\infty$?", "background": "This is a stronger form of the Erdos-Tur\\'{a}n conjecture [28] (since establishing this for any function $g(N)\\to \\infty$ would imply a positive solution to [28]).", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1898, "problem_number": "EP-41", "title": "Erdős Problem #41", "statement": "Let $A\\subset\\mathbb{N}$ be an infinite set such that the triple sums $a+b+c$ are all distinct for $a,b,c\\in A$ (aside from the trivial coincidences). Is it true that $ \\liminf \\frac{\\lvert A\\cap \\{1,\\ldots,N\\}\\rvert}{N^{1/3}}=0? $ ", "background": "Erdos proved that if the pairwise sums $a+b$ are all distinct aside from the trivial coincidences then $ \\liminf \\frac{\\lvert A\\cap \\{1,\\ldots,N\\}\\rvert}{N^{1/2}}=0. $ This is discussed in problem C11 of Guy's collection \\cite{Gu04}, in which Guy says Erdos offered \\$500 for the general problem of whether, for all $h\\geq 2$, $ \\liminf \\frac{\\lvert A\\cap \\{1,\\ldots,N\\}\\rvert}{N^{1/h}}=0 $ whenever the sum of $h$ terms in $A$ are distinct. This was proved for $h=4$ by Nash \\cite{Na89} and for all even $h$ by Chen \\cite{Ch96b}.\nReferences\n\n\n[Ch96b] Chen, Sheng, A note on {$B_{2k}$} sequences. J. Number Theory (1996), 1--3.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Na89] Nash, John C. M., On {$B_4$}-sequences. Canad. Math. Bull. (1989), 446--449.", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1899, "problem_number": "EP-42", "title": "Erdős Problem #42", "statement": "Let $M\\geq 1$ and $N$ be sufficiently large in terms of $M$. Is it true that for every Sidon set $A\\subset \\{1,\\ldots,N\\}$ there is another Sidon set $B\\subset \\{1,\\ldots,N\\}$ of size $M$ such that $(A-A)\\cap(B-B)=\\{0\\}$?", "background": "", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1900, "problem_number": "EP-43", "title": "Erdős Problem #43", "statement": "If $A,B\\subset \\{1,\\ldots,N\\}$ are two Sidon sets such that $(A-A)\\cap(B-B)=\\{0\\}$ then is it true that $ \\binom{\\lvert A\\rvert}{2}+\\binom{\\lvert B\\rvert}{2}\\leq\\binom{f(N)}{2}+O(1), $ where $f(N)$ is the maximum possible size of a Sidon set in $\\{1,\\ldots,N\\}$? If $\\lvert A\\rvert=\\lvert B\\rvert$ then can this bound be improved to $ \\binom{\\lvert A\\rvert}{2}+\\binom{\\lvert B\\rvert}{2}\\leq (1-c+o(1))\\binom{f(N)}{2} $ for some constant $c>0$?", "background": "Since it is known that $f(N)\\sim \\sqrt{N}$ (see [30]) the latter question is equivalent to asking whether, if $\\lvert A\\rvert=\\lvert B\\rvert$, $ \\lvert A\\rvert \\leq \\left(\\frac{1}{\\sqrt{2}}-c+o(1)\\right)\\sqrt{N} $ for some constant $c>0$. In the comments Tao has given a proof of this upper bound without the $-c$.\nIn the comments Barreto has given a negative answer to the second question: for infinitely many $N$ there exist Sidon sets $A,B\\subset \\{1,\\ldots,N\\}$ with $\\lvert A\\rvert=\\lvert B\\rvert$ and $(A-A)\\cap (B-B)=\\{0\\}$ and $ \\binom{\\lvert A\\rvert}{2}+\\binom{\\lvert B\\rvert}{2}\\geq (1-o(1))\\binom{f(N)}{2}. $ ", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1901, "problem_number": "EP-44", "title": "Erdős Problem #44", "statement": "Let $N\\geq 1$ and $A\\subset \\{1,\\ldots,N\\}$ be a Sidon set. Is it true that, for any $\\epsilon>0$, there exist $M$ and $B\\subset \\{N+1,\\ldots,M\\}$ (which may depend on $N,A,\\epsilon$) such that $A\\cup B\\subset \\{1,\\ldots,M\\}$ is a Sidon set of size at least $(1-\\epsilon)M^{1/2}$?", "background": "See also [329] and [707] (indeed a positive solution to [707] implies a positive solution to this problem, which in turn implies a positive solution to [329]).\nThis is discussed in problem C9 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1902, "problem_number": "EP-50", "title": "Erdős Problem #50", "statement": "Schoenberg proved that for every $c\\in [0,1]$ the density of $ \\{ n\\in \\mathbb{N} : \\phi(n)0$ $ \\max( \\lvert A+A\\rvert,\\lvert AA\\rvert)\\gg_\\epsilon \\lvert A\\rvert^{2-\\epsilon}? $ ", "background": "The sum-product problem. Erdos and Szemer\\'{e}di \\cite{ErSz83} proved a lower bound of $\\lvert A\\rvert^{1+c}$ for some constant $c>0$, and an upper bound of $ \\lvert A\\rvert^2 \\exp\\left(-c\\frac{\\log\\lvert A\\rvert}{\\log\\log \\lvert A\\rvert}\\right) $ for some constant $c>0$. The lower bound has been improved a number of times. The current record is $ \\max( \\lvert A+A\\rvert,\\lvert AA\\rvert)\\gg\\lvert A\\rvert^{\\frac{1270}{951}-o(1)} $ due to Bloom \\cite{Bl25} (note $1270/951=1.33543\\cdots$). A complete history of sum-product bounds can be found at this webpage.\nThere is likely nothing special about the integers in this question, and indeed Erdos and Szemer\\'{e}di also ask a similar question about finite sets of real or complex numbers. The current best bound for sets of reals is the same bound of Bloom above. The best bound for complex numbers is $ \\max( \\lvert A+A\\rvert,\\lvert AA\\rvert)\\gg\\lvert A\\rvert^{\\frac{4}{3}+c} $ for some absolute constant $c>0$, due to Basit and Lund \\cite{BaLu19}.\nOne can in general ask this question in any setting where addition and multiplication are defined (once one avoids any trivial obstructions such as zero divisors or finite subfields). For example, it makes sense for subsets of finite fields. The current record is that there exists $c>0$ such that if $A\\subseteq \\mathbb{F}_p$ with $\\lvert A\\rvert \\mathrm{ex}(n;C_4)$ edges contain $\\gg n^{1/2}$ many copies of $C_4$?", "background": "Conjectured by Erdos and Simonovits, who could not even prove that at least $2$ copies of $C_4$ are guaranteed.\nThe behaviour of $\\mathrm{ex}(n;C_4)$ is the subject of [765].\nHe, Ma, and Yang \\cite{HeMaYa21} have proved this conjecture when $n=q^2+q+1$ for some even integer $q$.\nReferences\n\n\n[HeMaYa21] He, J. and Ma, J. and Yang, T., Some extremal results on 4-cycles. Journal of Combinatorial Theory B (2021).", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1906, "problem_number": "EP-61", "title": "Erdős Problem #61", "statement": "For any graph $H$ is there some $c=c(H)>0$ such that every graph $G$ on $n$ vertices that does not contain $H$ as an induced subgraph contains either a complete graph or independent set on $\\geq n^c$ vertices?", "background": "Conjectured by Erdos and Hajnal \\cite{ErHa89}, who proved that a complete graph or independent set must exist on $ \\geq \\exp(c_H\\sqrt{\\log n}) $ many vertices, where $c_H>0$ is some constant. This was improved by Buci\\'{c}, Nguyen, Scott, and Seymour \\cite{BNSS23} to $ \\geq \\exp(c_H\\sqrt{\\log n\\log\\log n}). $ See also the entry in the graphs problem collection.\nReferences\n\n\n[BNSS23] Buci\\'C, M. and Nguyen, T. and Scott, A. and Seymour, P., A loglog step towards Erdos-Hajnal. arXiv:2301.10147 (2023).\n\n[ErHa89] Erdos, P. and Hajnal, A., Ramsey-type theorems. Discrete Appl. Math. (1989), 37-52.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1907, "problem_number": "EP-62", "title": "Erdős Problem #62", "statement": "If $G_1,G_2$ are two graphs with chromatic number $\\aleph_1$ then must there exist a graph $G$ whose chromatic number is $4$ (or even $\\aleph_0$) which is a subgraph of both $G_1$ and $G_2$?", "background": "Erdos also asked \\cite{Er87} about finding a common subgraph $H$ (with chromatic number either $4$ or $\\aleph_0$) in any finite collection of graphs with chromatic number $\\aleph_1$.\nEvery graph with chromatic number $\\aleph_1$ contains all sufficiently large odd cycles (which have chromatic number $3$), see [594]. This was proved by Erdos, Hajnal, and Shelah \\cite{EHS74}. Erdos wrote \\cite{Er87} that 'probably' every graph with chromatic number $\\aleph_1$ contains as subgraphs all graphs with chromatic number $4$ with sufficiently large girth.\nReferences\n\n\n[EHS74] Erdos, P. and Hajnal, A. and Shelah, S., On some general properties of chromatic numbers. Topics in topology (Proc. Colloq., Keszthely, 1972) (1974), 243-255.\n\n[Er87] Erdos, P., Some problems on finite and infinite graphs. Logic and combinatorics (Arcata, Calif., 1985) (1987), 223-228.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1908, "problem_number": "EP-65", "title": "Erdős Problem #65", "statement": "Let $G$ be a graph with $n$ vertices and $kn$ edges, and $a_10$.\nIt is open even for $f(n)=\\sqrt{n}$. Erdos offered \\$500 for a proof but only \\$250 for a counterexample. This fails (even with $f(n)\\gg n$) if the graph has chromatic number $\\aleph_1$ (see [111]).\nReferences\n\n\n[EHS82] Erdos, P. and Hajnal, A. and Szemer\\'{e}di, E., On almost bipartite large chromatic graphs. Theory and practice of combinatorics (1982), 117-123.\n\n[Ro82] R\"{o}dl, Vojt\\vEch, Nearly bipartite graphs with large chromatic number. Combinatorica (1982), 377-383.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1913, "problem_number": "EP-75", "title": "Erdős Problem #75", "statement": "Is there a graph of chromatic number $\\aleph_1$ such that for all $\\epsilon>0$ if $n$ is sufficiently large and $H$ is a subgraph on $n$ vertices then $H$ contains an independent set of size $>n^{1-\\epsilon}$?", "background": "Conjectured by Erdos, Hajnal, and Szemer\\'{e}di \\cite{EHS82}. In \\cite{Er95d} Erdos suggests this may even be true with an independent set of size $\\gg n$.\nSee also [750].\nReferences\n\n\n[EHS82] Erdos, P. and Hajnal, A. and Szemer\\'{e}di, E., On almost bipartite large chromatic graphs. Theory and practice of combinatorics (1982), 117-123.\n\n[Er95d] Erdos, Paul, On some problems in combinatorial set theory. Publ. Inst. Math. (Beograd) (N.S.) (1995), 61-65.", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1914, "problem_number": "EP-77", "title": "Erdős Problem #77", "statement": "If $R(k)$ is the Ramsey number for $K_k$, the minimal $n$ such that every $2$-colouring of the edges of $K_n$ contains a monochromatic copy of $K_k$, then find the value of $ \\lim_{k\\to \\infty}R(k)^{1/k}. $ ", "background": "Erdos offered \\$100 for just a proof of the existence of this constant, without determining its value. He also offered \\$1000 for a proof that the limit does not exist, but says 'this is really a joke as [it] certainly exists'. (In \\cite{Er88} he raises this prize to \\$10000). Erdos proved $ \\sqrt{2}\\leq \\liminf_{k\\to \\infty}R(k)^{1/k}\\leq \\limsup_{k\\to \\infty}R(k)^{1/k}\\leq 4. $ The upper bound has been improved to $4-\\tfrac{1}{128}$ by Campos, Griffiths, Morris, and Sahasrabudhe \\cite{CGMS23}. This was improved to $3.7992\\cdots$ by Gupta, Ndiaye, Norin, and Wei \\cite{GNNW24}.\nA shorter and simpler proof of an upper bound of the strength $4-c$ for some constant $c>0$ (and a generalisation to the case of more than two colours) was given by Balister, Bollob\\'{a}s, Campos, Griffiths, Hurley, Morris, Sahasrabudhe, and Tiba \\cite{BBCGHMST24}.\nIn \\cite{Er93} Erdos writes 'I have no idea what the value of $\\lim R(k)^{1/k}$ should be, perhaps it is $2$ but we have no real evidence for this.'\nThis problem is #3 in Ramsey Theory in the graphs problem collection.\nSee also [1029] for a problem concerning a lower bound for $R(k)$ and discussion of lower bounds in general.\nA famous quote of Erdos concerns the difficulty of finding exact values for $R(k)$. This is often repeated in the words of Spencer, who phrased it as an alien attacking race. The earliest such quote in a paper of Erdos I have found is in \\cite{Er93}, where he writes:\n'Sometime ago, I made the following joke. If an evil spirit would appear and say \"unless you give me the value of $R(5)$ within a year, I will exterminate humanity\", then our best bet would be perhaps to get all our computers working on $R(5)$ and we probably would get its value in a year.\nIf he would ask for $R(6)$, the best strategy probably would be to destroy it before it can destroy us. If we would be so clever that we could give the answer by mathematics, we would just tell him: \"if you try to do something you will see what will happent to you...\". I think we are strong enugh now and the only evil spirit we have to feel is the one which is in ourselves (quoting somebody: I have seen the enemy and them are us). Now enough of the idle talk and back to Mathematics.'\nReferences\n\n\n[BBCGHMST24] Balister, P. and Bollob\\'{a}s, B. and Campos, M. and Griffiths, S. and Hurley, E.\nand Morris, R. and Sahasrabudhe, J. and Tiba, M., Upper bounds for multicolour Ramsey numbers. arXiv:2410.17197 (2024).\n\n[CGMS23] Campos, Marcelo and Griffiths, Simon and Morris, Robert and Sahasrabudhe, Julian, An exponential improvement for diagonal Ramsey. arXiv:2303.09521 (2023).\n\n[Er88] Erdos, P, Problems and results in combinatorial analysis and graph theory. Discrete Math. (1988), 81-92.\n\n[Er93] Erdos, Paul, Some of my favorite solved and unsolved problems in graph\ntheory. Quaestiones Math. (1993), 333-350.\n\n[GNNW24] Gupta, P. and Ndiaye, N. and Norin, S. and Wei, L., Optimizing the CGMS upper bound on Ramsey numbers. arXiv:2407.19026 (2024).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1915, "problem_number": "EP-78", "title": "Erdős Problem #78", "statement": "Give a constructive proof that $R(k)>C^k$ for some constant $C>1$.", "background": "Erdos gave a simple probabilistic proof that $R(k) \\gg k2^{k/2}$.\nEquivalently, this question asks for an explicit construction of a graph on $n$ vertices which does not contain any clique or independent set of size $\\geq c\\log n$ for some constant $c>0$.\nIn \\cite{Er69b} Erdos asks for even a construction whose largest clique or independent set has size $o(n^{1/2})$, which is now known.\nCohen \\cite{Co15} (see the introduction for further history) constructed a graph on $n$ vertices which does not contain any clique or independent set of size $ \\geq 2^{(\\log\\log n)^{C}} $ for some constant $C>0$. Li \\cite{Li23b} has recently improved this to $ \\geq (\\log n)^{C} $ for some constant $C>0$.\nThis problem is #4 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[Co15] Gil Cohen, Two-Source Dispersers for Polylogarithmic Entropy and Improved Ramsey Graphs. Electronic Colloquium on Computational Complexity (2015).\n\n[Er69b] Erdos, P., Problems and results in chromatic graph theory. Proof Techniques in Graph Theory (Proc. Second Ann\nArbor Graph Theory Conf., Ann Arbor, Mich.,\n1968) (1969), 27-35.\n\n[Li23b] Li, X., Two Source Extractors for Asymptotically Optimal Entropy, and (Many) More. arXiv:2303.06802 (2023).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1916, "problem_number": "EP-80", "title": "Erdős Problem #80", "statement": "Let $c>0$ and let $f_c(n)$ be the maximal $m$ such that every graph $G$ with $n$ vertices and at least $cn^2$ edges, where each edge is contained in at least one triangle, must contain a book of size $m$, that is, an edge shared by at least $m$ different triangles.\nEstimate $f_c(n)$. In particular, is it true that $f_c(n)>n^{\\epsilon}$ for some $\\epsilon>0$? Or $f_c(n)\\gg \\log n$?", "background": "A problem of Erdos and Rothschild. Alon and Trotter showed that, provided $c<1/4$, $f_c(n)\\ll_c n^{1/2}$. Szemer\\'{e}di observed that his regularity lemma implies that $f_c(n)\\to \\infty$.\nEdwards (unpublished) and Khadziivanov and Nikiforov \\cite{KhNi79} proved independently that $f_c(n) \\geq n/6$ when $c>1/4$ (see [905]).\nFox and Loh \\cite{FoLo12} proved that $ f_c(n) \\leq n^{O(1/\\log\\log n)} $ for all $c<1/4$, disproving the first conjecture of Erdos.\nThe best known lower bounds for $f_c(n)$ are those from Szemer\\'{e}di's regularity lemma, and as such remain very poor.\nSee also [600] and the entry in the graphs problem collection.\nReferences\n\n\n[FoLo12] Fox, Jacob and Loh, Po-Shen, On a problem of Erdos and {R}othschild on edges in\ntriangles. Combinatorica (2012), 619--628.\n\n[KhNi79] Had\\v ziivanov, N. G. and Nikiforov, S. V., Solution of a problem of {P}. Erdos about the maximum\nnumber of triangles with a common edge in a graph. C. R. Acad. Bulgare Sci. (1979), 1315--1318.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1917, "problem_number": "EP-81", "title": "Erdős Problem #81", "statement": "Let $G$ be a chordal graph on $n$ vertices - that is, $G$ has no induced cycles of length greater than $3$. Can the edges of $G$ be partitioned into $n^2/6+O(n)$ many cliques?", "background": "Asked by Erdos, Ordman, and Zalcstein \\cite{EOZ93}, who proved an upper bound of $(1/4-\\epsilon)n^2$ many cliques (for some very small $\\epsilon>0$). The example of all edges between a complete graph on $n/3$ vertices and an empty graph on $2n/3$ vertices show that $n^2/6+O(n)$ is sometimes necessary.\nA split graph is one where the vertices can be split into a clique and an independent set. Every split graph is chordal. Chen, Erdos, and Ordman \\cite{CEO94} have shown that any split graph can be partitioned into $\\frac{3}{16}n^2+O(n)$ many cliques.\nSee also [1017].\nReferences\n\n\n[CEO94] Chen, Guan-Tao and Erdos, Paul and Ordman, Edward T., Clique partitions of split graphs. Combinatorics, graph theory, algorithms and applications\n(Beijing, 1993) (1994), 21-30.\n\n[EOZ93] Erdos, Paul and Ordman, Edward T. and Zalcstein, Yechezkel, Clique partitions of chordal graphs. Combin. Probab. Comput. (1993), 409-415.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1918, "problem_number": "EP-82", "title": "Erdős Problem #82", "statement": "Let $F(n)$ be maximal such that every graph on $n$ vertices contains a regular induced subgraph on at least $F(n)$ vertices. Prove that $F(n)/\\log n\\to \\infty$.", "background": "Conjectured by Erdos, Fajtlowicz, and Stanton. It is known that $F(5)=3$ and $F(7)=4$.\nRamsey's theorem implies that $F(n)\\gg \\log n$. Bollob\\'{a}s observed that $F(n)\\ll n^{1/2+o(1)}$. Alon, Krivelevich, and Sudakov \\cite{AKS07} have improved this to $n^{1/2}(\\log n)^{O(1)}$.\nIn \\cite{Er93} Erdos asks whether, if $t(n)$ is the largest trivial (either empty or complete) subgraph which a graph on $n$ vertices must contain (so that $t(n) \\gg \\log n$ by Ramsey's theorem), then is it true that $ F(n)-t(n)\\to \\infty? $ Equivalently, and in analogue with the definition of Ramsey numbers, one can define $G(n)$ to be the minimal $m$ such that every graph on $m$ vertices contains a regular induced subgraph on at least $n$ vertices. This problem can be rephrased as asking whether $G(n) \\leq 2^{o(n)}$.\nFajtlowicz, McColgan, Reid, and Staton \\cite{FMRS95} showed that $G(1)=1$, $G(2)=2$, $G(3)=5$, $G(4)=7$, and $G(5)\\geq 12$. Boris Alexeev and Brendan McKay (see the comments and this site) have computed $G(5)=17$, $G(6)\\geq 21$, and $G(7)\\geq 29$.\nSee also [1031] for another question regarding induced regular subgraphs.\nReferences\n\n\n[AKS07] Alon, N. and Krivelevich, M. and Sudakov, B., Large nearly regular induced subgraphs. arXiv:0710.2106 (2007).\n\n[Er93] Erdos, Paul, Some of my favorite solved and unsolved problems in graph\ntheory. Quaestiones Math. (1993), 333-350.\n\n[FMRS95] No reference found.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1919, "problem_number": "EP-84", "title": "Erdős Problem #84", "statement": "The cycle set of a graph $G$ on $n$ vertices is a set $A\\subseteq \\{3,\\ldots,n\\}$ such that there is a cycle in $G$ of length $\\ell$ if and only if $\\ell \\in A$. Let $f(n)$ count the number of possible such $A$.\nProve that $f(n)=o(2^n)$.\nProve that $f(n)/2^{n/2}\\to \\infty$.", "background": "Conjectured by Erdos and Faudree, who showed that $2^{n/2}0$, and wrote it is 'perhaps not hopeless' to determine $f(n)$ exactly. Brass, Harborth, and Nienborg \\cite{BHN95} improved this to $ f(n) \\geq \\left(\\frac{1}{2}+\\frac{c}{\\sqrt{n}}\\right)n2^{n-1} $ for some constant $c>0$.\nBalogh, Hu, Lidicky, and Liu \\cite{BHLL14} proved that $f(n)\\leq 0.6068 n2^{n-1}$. This was improved to $\\leq 0.60318 n2^{n-1}$ by Baber \\cite{Ba12b}.\nA similar question can be asked for other even cycles.\nSee also [666] and the entry in the graphs problem collection.\nReferences\n\n\n[BHLL14] Balogh, J\\'{o}zsef and Hu, Ping and Lidick\\'{y}, Bernard and Liu, Hong, Upper bounds on the size of 4- and 6-cycle-free subgraphs of the hypercube. European J. Combin. (2014), 75-85.\n\n[BHN95] Brass, Peter and Harborth, Heiko and Nienborg, Hauke, On the maximum number of edges in a {$C_4$}-free subgraph of\n{$Q_n$}. J. Graph Theory (1995), 17--23.\n\n[Ba12b] R. Baber, Tur\\'{a}n densities of hypercubes. arXiv:1201.3587 (2012).\n\n[Er91] Erd\"{o}s, P., Problems and results in combinatorial analysis and combinatorial number theory. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988) (1991), 397-406.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1921, "problem_number": "EP-87", "title": "Erdős Problem #87", "statement": "Let $\\epsilon >0$. Is it true that, if $k$ is sufficiently large, then $ R(G)>(1-\\epsilon)^kR(k) $ for every graph $G$ with chromatic number $\\chi(G)=k$?\nEven stronger, is there some $c>0$ such that, for all large $k$, $R(G)>cR(k)$ for every graph $G$ with chromatic number $\\chi(G)=k$?", "background": "Erdos originally conjectured that $R(G)\\geq R(k)$, which is trivial for $k=3$, but fails already for $k=4$, as Faudree and McKay \\cite{FaMc93} showed that $R(W)=17$ for the pentagonal wheel $W$.\nSince $R(k)\\leq 4^k$ this is trivial for $\\epsilon\\geq 3/4$. Yuval Wigderson points out that $R(G)\\gg 2^{k/2}$ for any $G$ with chromatic number $k$ (via a random colouring), which asymptotically matches the best-known lower bounds for $R(k)$.\nThis problem is #12 and #13 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[FaMc93] Faudree, R. J. and McKay, B., A conjecture of Erdos and the Ramsey number $r(W_6)$. J. Combinatorial Math. and Combinatorial Computing (1993), 23-31.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1922, "problem_number": "EP-89", "title": "Erdős Problem #89", "statement": "Does every set of $n$ distinct points in $\\mathbb{R}^2$ determine $\\gg n/\\sqrt{\\log n}$ many distinct distances?", "background": "A $\\sqrt{n}\\times\\sqrt{n}$ integer grid shows that this would be the best possible. Nearly solved by Guth and Katz \\cite{GuKa15} who proved that there are always $\\gg n/\\log n$ many distinct distances.\nA stronger form (see [604]) may be true: is there a single point which determines $\\gg n/\\sqrt{\\log n}$ distinct distances, or even $\\gg n$ many such points, or even that this is true averaged over all points - for example, if $d(x)$ counts the number of distinct distances from $x$ then in \\cite{Er75f} Erdos conjectured $ \\sum_{x\\in A}d(x) \\gg \\frac{n^2}{\\sqrt{\\log n}}, $ where $A\\subset \\mathbb{R}^2$ is any set of $n$ points.\nSee also [661], and [1083] for the generalisation to higher dimensions.\nReferences\n\n\n[Er75f] Erdos, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108.\n\n[GuKa15] Guth, Larry and Katz, Nets Hawk, On the Erdos distinct distances problem in the plane. Ann. of Math. (2) (2015), 155-190.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1923, "problem_number": "EP-90", "title": "Erdős Problem #90", "statement": "Does every set of $n$ distinct points in $\\mathbb{R}^2$ contain at most $n^{1+O(1/\\log\\log n)}$ many pairs which are distance 1 apart?", "background": "The unit distance problem. In \\cite{Er94b} Erdos dates this conjecture to 1946. In \\cite{Er82e} he offers \\$300 for the upper bound $n^{1+o(1)}$.\nThis would be the best possible, as is shown by a set of lattice points. It is easy to show that there are $O(n^{3/2})$ many such pairs. The best known upper bound is $O(n^{4/3})$, due to Spencer, Szemer\\'{e}di, and Trotter \\cite{SST84}. In \\cite{Er83c} and \\cite{Er85} Erdos offers \\$250 for an upper bound of the form $n^{1+o(1)}$.\nPart of the difficulty of this problem is explained by a result of Valtr (see \\cite{Sz16}), who constructed a metric on $\\mathbb{R}^2$ and a set of $n$ points with $\\gg n^{4/3}$ unit distance pairs (with respect to this metric). The methods of the upper bound proof of Spencer, Szemer\\'{e}di, and Trotter \\cite{SST84} generalise to include this metric. Therefore to prove an upper bound better than $n^{4/3}$ some special feature of the Euclidean metric must be exploited.\nSee a survey by Szemer\\'{e}di \\cite{Sz16} for further background and related results.\nSee also [92], [96], [605], and [956]. The higher dimensional generalisation is [1085].\nReferences\n\n\n[Er82e] Erdos, Paul, Some of my favourite problems which recently have been solved. (1982), 59--79.\n\n[Er83c] Erdos, Paul, Combinatorial problems in geometry. Math. Chronicle (1983), 35-54.\n\n[Er85] Erdos, P., Problems and results in combinatorial geometry. Discrete geometry and convexity (New York, 1982) (1985), 1-11.\n\n[Er94b] Erdos, Paul, Some problems in number theory, combinatorics and combinatorial geometry. Math. Pannon. (1994), 261-269.\n\n[SST84] Spencer, J. and Szemer\\'{e}di, E. and Trotter, Jr., W., Unit distances in the Euclidean plane. Graph theory and combinatorics (Cambridge, 1983) (1984), 293-303.\n\n[Sz16] Szemer\\'{e}di, Endre, Erdos's unit distance problem. Open problems in mathematics (2016), 459-477.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1924, "problem_number": "EP-91", "title": "Erdős Problem #91", "statement": "Let $n$ be a sufficently large integer. Suppose $A\\subset \\mathbb{R}^2$ has $\\lvert A\\rvert=n$ and minimises the number of distinct distances between points in $A$. Prove that there are at least two (and probably many) such $A$ which are non-similar.", "background": "For $n=3$ the equilateral triangle is the only such set. For $n=4$ the square or two equilateral triangles sharing an edge give two non-similar examples.\nFor $n=5$ the regular pentagon is the unique such set (which has two distinct distances). Erdos mysteriously remarks in \\cite{Er90} this was proved by 'a colleague'. (In \\cite{Er87b} this is described as 'a colleague from Zagreb (unfortunately I do not have his letter)'.) A published proof of this fact is provided by Kov\\'{a}cs \\cite{Ko24c}.\nIn \\cite{Er87b} Erdos says that there are at least two non-similar examples for $6\\leq n\\leq 9$.\nThe minimal possible number of distinct distances is the subject of [89].\nReferences\n\n\n[Er87b] Erdos, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Si\\'{o}fok, 1985) (1987), 167-177.\n\n[Er90] Erdos, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdos (1990), 467-478.\n\n[Ko24c] Z. Kov\\'{a}cs, A note on Erdos's mysterious remark. arXiv:2412.05190 (2024).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1925, "problem_number": "EP-92", "title": "Erdős Problem #92", "statement": "Let $f(n)$ be maximal such that there exists a set $A$ of $n$ points in $\\mathbb{R}^2$ in which every $x\\in A$ has at least $f(n)$ points in $A$ equidistant from $x$.\nIs it true that $f(n)\\leq n^{o(1)}$? Or even $f(n) < n^{O(1/\\log\\log n)}$?", "background": "This is a stronger form of the unit distance conjecture (see [90]).\nThe set of lattice points imply $f(n) > n^{c/\\log\\log n}$ for some constant $c>0$. Erdos offered \\$500 for a proof that $f(n) \\leq n^{o(1)}$ but only \\$100 for a counterexample. This latter prize is downgraded to \\$50 in \\cite{ErFi97}.\nIt is trivial that $f(n) \\ll n^{1/2}$. A result of Pach and Sharir (Theorem 4 of \\cite{PaSh92}) implies $f(n) \\ll n^{2/5}$. Hunter has observed that the circle-point incidence bound of Janzer, Janzer, Methuku, and Tardos \\cite{JJMT24} implies $ f(n) \\ll n^{4/11}. $ Fishburn (personal communication to Erdos, later published in \\cite{ErFi97}) proved that $6$ is the smallest $n$ such that $f(n)=3$ and $8$ is the smallest $n$ such that $f(n)=4$, and suggested that the lattice points may not be best example.\nSee also [754].\nReferences\n\n\n[ErFi97] Erdos, Paul and Fishburn, Peter, Minimum planar sets with maximum equidistance counts. Comput. Geom. (1997), 207--218.\n\n[JJMT24] B. Janzer, O. Janzer, A. Methuku, and G. Tardos, Tight bounds for intersection-reverse sequences, edge-ordered graphs\nand applications. arXiv:2411.07188 (2024).\n\n[PaSh92] Pach, J\\'anos and Sharir, Micha, Repeated angles in the plane and related problems. J. Combin. Theory Ser. A (1992), 12--22.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1926, "problem_number": "EP-96", "title": "Erdős Problem #96", "statement": "If $n$ points in $\\mathbb{R}^2$ form a convex polygon then there are $O(n)$ many pairs which are distance $1$ apart.", "background": "Conjectured by Erdos and Moser. In \\cite{Er92e} Erdos credits the conjecture that the true upper bound is $2n$ to himself and Fishburn. F\"{u}redi \\cite{Fu90} proved an upper bound of $O(n\\log n)$. A short proof of this bound was given by Brass and Pach \\cite{BrPa01}. The best known upper bound is $ \\leq n\\log_2n+4n, $ due to Aggarwal \\cite{Ag15}.\nEdelsbrunner and Hajnal \\cite{EdHa91} have constructed $n$ such points with $2n-7$ pairs distance $1$ apart. (This disproved an early stronger conjecture of Erdos and Moser, that the true answer was $\\frac{5}{3}n+O(1)$.)\nA positive answer would follow from [97]. See also [90].\nIn \\cite{Er92e} Erdos makes the stronger conjecture that, if $g(x)$ counts the largest number of points equidistant from $x$ in $A$, then $ \\sum_{x\\in A}g(x)< 4n. $ He notes that the example of Edelsbrunner and Hajnal shows that $\\sum_{x\\in A}g(x)>4n-O(1)$ is possible.\nReferences\n\n\n[Ag15] Aggarwal, Amol, On unit distances in a convex polygon. Discrete Math. (2015), 88-92.\n\n[BrPa01] Brass , Peter and Pach, J\\'{a}nos, The maximum number of times the same distance can occur among\nthe vertices of a convex {$n$}-gon is {$O(n\\log n)$}. J. Combin. Theory Ser. A (2001), 178-179.\n\n[EdHa91] Edelsbrunner, Herbert and Hajnal, P\\'{e}ter, A lower bound on the number of unit distances between the\nvertices of a convex polygon. J. Combin. Theory Ser. A (1991), 312-316.\n\n[Er92e] Erdos, P\\'{a}l, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48.\n\n[Fu90] F\"{u}redi, Zolt\\'{a}n, The maximum number of unit distances in a convex {$n$}-gon. J. Combin. Theory Ser. A (1990), 316-320.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1927, "problem_number": "EP-98", "title": "Erdős Problem #98", "statement": "Let $h(n)$ be such that any $n$ points in $\\mathbb{R}^2$, with no three on a line and no four on a circle, determine at least $h(n)$ distinct distances. Does $h(n)/n\\to \\infty$?", "background": "Erdos could not even prove $h(n)\\geq n$. Pach has shown $h(n)0$.\nReferences\n\n\n[EFPR93] Erdos, Paul and F\"{u}redi, Zolt\\'{a}n and Pach, J\\'{a}nos and\nRuzsa, Imre Z., The grid revisited. Discrete Math. (1993), 189--196.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1928, "problem_number": "EP-99", "title": "Erdős Problem #99", "statement": "Let $A\\subseteq\\mathbb{R}^2$ be a set of $n$ points with minimum distance equal to 1, chosen to minimise the diameter of $A$. If $n$ is sufficiently large then must there be three points in $A$ which form an equilateral triangle of size 1?", "background": "Thue proved that the minimal such diameter is achieved (asymptotically) by the points in a triangular lattice intersected with a circle. In general Erdos believed such a set must have very large intersection with the triangular lattice (perhaps as many as $(1-o(1))n$).\nErdos \\cite{Er94b} wrote 'I could not prove it but felt that it should not be hard. To my great surprise both B. H. Sendov and M. Simonovits doubted the truth of this conjecture.' In \\cite{Er94b} he offers \\$100 for a counterexample but only \\$50 for a proof.\nThe stated problem is false for $n=4$, for example taking the points to be vertices of a square. The behaviour of such sets for small $n$ is explored by Bezdek and Fodor \\cite{BeFo99}.\nSee also [103].\nReferences\n\n\n[BeFo99] Bezdek, Andr\\'{a}s and Fodor, Ferenc, Minimal diameter of certain sets in the plane. J. Combin. Theory Ser. A (1999), 105-111.\n\n[Er94b] Erdos, Paul, Some problems in number theory, combinatorics and combinatorial geometry. Math. Pannon. (1994), 261-269.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1929, "problem_number": "EP-100", "title": "Erdős Problem #100", "statement": "Let $A$ be a set of $n$ points in $\\mathbb{R}^2$ such that all pairwise distances are at least $1$ and if two distinct distances differ then they differ by at least $1$. Is the diameter of $A$ $\\gg n$?", "background": "Perhaps the diameter is even $\\geq n-1$ for sufficiently large $n$. Piepmeyer has an example of $9$ such points with diameter $<5$. Kanold proved the diameter is $\\geq n^{3/4}$. The bounds on the distinct distance problem [89] proved by Guth and Katz \\cite{GuKa15} imply a lower bound of $\\gg n/\\log n$.\nReferences\n\n\n[GuKa15] Guth, Larry and Katz, Nets Hawk, On the Erdos distinct distances problem in the plane. Ann. of Math. (2) (2015), 155-190.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1930, "problem_number": "EP-101", "title": "Erdős Problem #101", "statement": "Given $n$ points in $\\mathbb{R}^2$, no five of which are on a line, the number of lines containing four points is $o(n^2)$.", "background": "There are examples of sets of $n$ points with $\\sim n^2/6$ many collinear triples and no four points on a line. Such constructions are given by Burr, Gr\"{u}nbaum, and Sloane \\cite{BGS74} and F\"{u}redi and Pal\\'{a}sti \\cite{FuPa84}.\nGr\"{u}nbaum \\cite{Gr76} constructed an example with $\\gg n^{3/2}$ such lines. Erdos speculated this may be the correct order of magnitude. This is false: Solymosi and Stojakovi\\'{c} \\cite{SoSt13} have constructed a set with no five on a line and at least $ n^{2-O(1/\\sqrt{\\log n})} $ many lines containing exactly four points.\nSee also [102] and [669]. A generalisation of this problem is asked in [588].\nThis problem is Problem 71 on Green's open problems list.\nReferences\n\n\n[BGS74] Burr, Stefan A. and Gr\"{u}nbaum, Branko and Sloane, N. J. A., The orchard problem. Geometriae Dedicata (1974), 397-424.\n\n[FuPa84] F\"{u}redi, Z. and Pal\\'{a}sti, I., Arrangements of lines with a large number of triangles. Proc. Amer. Math. Soc. (1984), 561-566.\n\n[Gr76] Gr\"{u}nbaum, Branko, New views on some old questions of combinatorial geometry. Colloquio Internazionale sulle Teorie Combinatorie\n(Roma, 1973), Tomo I (1976), 451-468.\n\n[SoSt13] Solymosi, J\\'{o}zsef and Stojakovi\\'C, Milo\\vS, Many collinear {$k$}-tuples with no {$k+1$} collinear points. Discrete Comput. Geom. (2013), 811-820.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1931, "problem_number": "EP-102", "title": "Erdős Problem #102", "statement": "Let $c>0$ and $h_c(n)$ be such that for any $n$ points in $\\mathbb{R}^2$ such that there are $\\geq cn^2$ lines each containing more than three points, there must be some line containing $h_c(n)$ many points. Estimate $h_c(n)$. Is it true that, for fixed $c>0$, we have $h_c(n)\\to \\infty$?", "background": "A problem of Erdos and Purdy. It is not even known if $h_c(n)\\geq 5$ (see [101]).\nIt is easy to see that $h_c(n) \\ll_c n^{1/2}$, and Erdos at one point \\cite{Er95} suggested that perhaps a similar lower bound $h_c(n)\\gg_c n^{1/2}$ holds. Zach Hunter has pointed out that this is false, even replacing $>3$ points on each line with $>k$ points: consider the set of points in $\\{1,\\ldots,m\\}^d$ where $n\\approx m^d$. These intersect any line in $\\ll_d n^{1/d}$ points, and have $\\gg_d n^2$ many pairs of points each of which determine a line with at least $k$ points. This is a construction in $\\mathbb{R}^d$, but a random projection into $\\mathbb{R}^2$ preserves the relevant properties.\nThis construction shows that $h_c(n) \\ll n^{1/\\log(1/c)}$.\nReferences\n\n\n[Er95] Erdos, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1932, "problem_number": "EP-103", "title": "Erdős Problem #103", "statement": "Let $h(n)$ count the number of incongruent sets of $n$ points in $\\mathbb{R}^2$ which minimise the diameter subject to the constraint that $d(x,y)\\geq 1$ for all points $x\neq y$. Is it true that $h(n)\\to \\infty$?", "background": "It is not even known whether $h(n)\\geq 2$ for all large $n$.\nSee also [99].\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1933, "problem_number": "EP-104", "title": "Erdős Problem #104", "statement": "Given $n$ points in $\\mathbb{R}^2$ the number of distinct unit circles containing at least three points is $o(n^2)$.", "background": "In \\cite{Er81d} Erdos proved that $\\gg n$ many circles is possible, and that there cannot be more than $O(n^2)$ many circles. The argument is very simple: every pair of points determines at most $2$ unit circles, and the claimed bound follows from double counting. Erdos claims in a number of places this produces the upper bound $n(n-1)$, but Harborth and Mengerson \\cite{HaMe86} note that in fact this delivers an upper bound of $\\frac{n(n-1)}{3}$.\nElekes \\cite{El84} has a simple construction of a set with $\\gg n^{3/2}$ such circles. This may be the correct order of magnitude.\nIn \\cite{Er75h} and \\cite{Er92e} Erdos also asks how many such unit circles there must be if the points are in general position.\nIn \\cite{Er92e} Erdos offered £100 for a proof or disproof that the answer is $O(n^{3/2})$.\nThe maximal number of unit circles achieved by $n$ points is A003829 in the OEIS.\nSee also [506] and [831].\nReferences\n\n\n[El84] Elekes, G., {$n$} points in the plane can determine $n^{3/2}$ unit\ncircles. Combinatorica (1984), 131.\n\n[Er75h] Erdos, P., Some problems on elementary geometry. Austral. Math. Soc. Gaz. (1975), 2-3.\n\n[Er81d] Erdos, P., Some applications of graph theory and combinatorial methods to number theory and geometry. Algebraic methods in graph theory, Vol. I, II (Szeged, 1978) (1981), 137-148.\n\n[Er92e] Erdos, P\\'{a}l, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48.\n\n[HaMe86] Harborth, Heiko and Mengersen, Ingrid, Point sets with many unit circles. Discrete Math. (1986), 193--197.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1934, "problem_number": "EP-108", "title": "Erdős Problem #108", "statement": "For every $r\\geq 4$ and $k\\geq 2$ is there some finite $f(k,r)$ such that every graph of chromatic number $\\geq f(k,r)$ contains a subgraph of girth $\\geq r$ and chromatic number $\\geq k$?", "background": "Conjectured by Erdos and Hajnal. R\"{o}dl \\cite{Ro77} has proved the $r=4$ case (see [923]). The infinite version (whether every graph of infinite chromatic number contains a subgraph of infinite chromatic number whose girth is $>k$) is also open.\nIn \\cite{Er79b} Erdos also asks whether $ \\lim_{k\\to \\infty}\\frac{f(k,r+1)}{f(k,r)}=\\infty. $ See also the entry in the graphs problem collection and [740] for the infinitary version.\nReferences\n\n\n[Er79b] Erdos, Paul, Problems and results in graph theory and combinatorial analysis. Graph theory and related topics (Proc. Conf., Univ. Waterloo, Waterloo, Ont., 1977) (1979), 153-163.\n\n[Ro77] R\"{o}dl, V., On the chromatic number of subgraphs of a given graph. Proc. Amer. Math. Soc. (1977), 370-371.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1935, "problem_number": "EP-111", "title": "Erdős Problem #111", "statement": "If $G$ is a graph let $h_G(n)$ be defined such that any subgraph of $G$ on $n$ vertices can be made bipartite after deleting at most $h_G(n)$ edges.\nWhat is the behaviour of $h_G(n)$? Is it true that $h_G(n)/n\\to \\infty$ for every graph $G$ with chromatic number $\\aleph_1$?", "background": "A problem of Erdos, Hajnal, and Szemer\\'{e}di \\cite{EHS82}. Every $G$ with chromatic number $\\aleph_1$ must have $h_G(n)\\gg n$ since $G$ must contain, for some $r$, $\\aleph_1$ many vertex disjoint odd cycles of length $2r+1$.\nOn the other hand, Erdos, Hajnal, and Szemer\\'{e}di proved that there is a $G$ with chromatic number $\\aleph_1$ such that $h_G(n)\\ll n^{3/2}$. In \\cite{Er81} Erdos conjectured that this can be improved to $\\ll n^{1+\\epsilon}$ for every $\\epsilon>0$.\nSee also [74].\nReferences\n\n\n[EHS82] Erdos, P. and Hajnal, A. and Szemer\\'{e}di, E., On almost bipartite large chromatic graphs. Theory and practice of combinatorics (1982), 117-123.\n\n[Er81] Erdos, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1936, "problem_number": "EP-112", "title": "Erdős Problem #112", "statement": "Let $k=k(n,m)$ be minimal such that any directed graph on $k$ vertices must contain either an independent set of size $n$ or a transitive tournament of size $m$. Determine $k(n,m)$.", "background": "A problem of Erdos and Rado \\cite{ErRa67}, who showed $k(n,m) \\ll_m n^{m-1}$, or more precisely, $ k(n,m) \\leq \\frac{2^{m-1}(n-1)^m+n-2}{2n-3}. $ Larson and Mitchell \\cite{LaMi97} improved the dependence on $m$, establishing in particular that $k(n,3)\\leq n^{2}$. Zach Hunter has observed that $ R(n,m) \\leq k(n,m)\\leq R(n,m,m), $ which in particular proves the upper bound $k(n,m)\\leq 3^{n+2m}$.\nSee also the entry in the graphs problem collection - on this site the problem replaces transitive tournament with directed path, but Zach Hunter and Raphael Steiner have a simple argument that proves, for this alternative definition, that $k(n,m)=(n-1)(m-1)$.\nReferences\n\n\n[ErRa67] Erdos, P. and Rado, R., Partition relations and transitivity domains of binary\nrelations. J. London Math. Soc. (1967), 624-633.\n\n[LaMi97] Larson, Jean A. and Mitchell, William J., On a problem of Erdos and Rado. Ann. Comb. (1997), 245-252.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1937, "problem_number": "EP-114", "title": "Erdős Problem #114", "statement": "If $p(z)\\in\\mathbb{C}[z]$ is a monic polynomial of degree $n$ then is the length of the curve $\\{ z\\in \\mathbb{C} : \\lvert p(z)\\rvert=1\\}$ maximised when $p(z)=z^n-1$?", "background": "A problem of Erdos, Herzog, and Piranian \\cite{EHP58}. It is also listed as Problem 4.10 in \\cite{Ha74}, where it is attributed to Erdos.\nLet the maximal length of such a curve be denoted by $f(n)$.\n{UL}\n{LI}The length of the curve when $p(z)=z^n-1$ is $2n+O(1)$, and hence the conjecture implies in particular that $f(n)=2n+O(1)$.{/LI}\n{LI}Dolzhenko \\cite{Do61} proved $f(n) \\leq 4\\pi n$, but few were aware of this work.{/LI}\n{LI}Pommerenke \\cite{Po61} proved $f(n)\\ll n^2$.{/LI}\n{LI}Borwein \\cite{Bo95} proved $f(n)\\ll n$ (Borwein was unaware of Dolzhenko's earlier work). The prize of \\$250 is reported by Borwein \\cite{Bo95}.{/LI}\n{LI}Eremenko and Hayman \\cite{ErHa99} proved the full conjecture when $n=2$, and $f(n)\\leq 9.173n$ for all $n$.{/LI}\n{LI}Danchenko \\cite{Da07} proved $f(n)\\leq 2\\pi n$.{/LI}\n{LI}Fryntov and Nazarov \\cite{FrNa09} proved that $z^n-1$ is a local maximiser, and solved this problem asymptotically, proving that $ f(n)\\leq 2n+O(n^{7/8}). $ {/LI}\n{LI} Tao \\cite{Ta25} has proved that $p(z)=z^n-1$ is the unique (up to rotation and translation) maximiser for all sufficiently large $n$.\n{/UL}\nErdos, Herzog, and Piranian \\cite{EHP58} also ask whether the length is at least $2\\pi$ if $\\{ z: \\lvert f(z)\\rvert<1\\}$ is connected (which $z^n$ shows is the best possible). This was proved by Pommerenke \\cite{Po59}.\nReferences\n\n\n[Bo95] Borwein, Peter, The arc length of the lemniscate {$\\{|p(z)|=1\\}$}. Proc. Amer. Math. Soc. (1995), 797--799.\n\n[Da07] Danchenko, V. I., The lengths of lemniscates. {V}ariations of rational\nfunctions. Mat. Sb. (2007), 51--58.\n\n[Do61] Dol\\v zenko, E. P., Some estimates concerning algebraic hypersurfaces and\nderivatives of rational functions. Dokl. Akad. Nauk SSSR (1961), 1287--1290.\n\n[EHP58] Erdos, P. and Herzog, F. and Piranian, G., Metric properties of polynomials. J. Analyse Math. (1958), 125-148.\n\n[ErHa99] Eremenko, Alexandre and Hayman, Walter, On the length of lemniscates. Michigan Math. J. (1999), 409--415.\n\n[FrNa09] Fryntov, Alexander and Nazarov, Fedor, New estimates for the length of the {E}rd\\H\nos-{H}erzog-{P}iranian lemniscate. (2009), 49--60.\n\n[Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155--180.\n\n[Po59] Pommerenke, Ch., On some problems by Erdos, Herzog and Piranian. Michigan Math. J. (1959), 221-225.\n\n[Po61] Pommerenke, Ch., On metric properties of complex polynomials. Michigan Math. J. (1961), 97-115.\n\n[Ta25] T. Tao, The maximal length of the Erdos-Herzog-Piranian leminscate length in high degree. arXiv:2512.12455 (2025).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1938, "problem_number": "EP-117", "title": "Erdős Problem #117", "statement": "Let $h(n)$ be minimal such that any group $G$ with the property that any subset of $>n$ elements contains some $x\neq y$ such that $xy=yx$ can be covered by at most $h(n)$ many Abelian subgroups.\nEstimate $h(n)$ as well as possible.", "background": "Pyber \\cite{Py87} has proved there exist constants $c_2>c_1>1$ such that $c_1^n0$ such that for infinitely many $n$ we have $M_n > n^c$?\nIs it true that there exists $c>0$ such that, for all large $n$, $ \\sum_{k\\leq n}M_k > n^{1+c}? $ ", "background": "This is Problem 4.1 in \\cite{Ha74} where it is attributed to Erdos.\nThe weaker conjecture that $\\limsup M_n=\\infty$ was proved by Wagner \\cite{Wa80}, who show that there is some $c>0$ with $M_n>(\\log n)^c$ infinitely often.\nThe second question was answered by Beck \\cite{Be91}, who proved that there exists some $c>0$ such that $ \\max_{n\\leq N} M_n > N^c. $ Erdos (e.g. see \\cite{Ha74}) gave a construction of a sequence with $M_n\\leq n+1$ for all $n$. Linden \\cite{Li77} improved this to give a sequence with $M_n\\ll n^{1-c}$ for some $c>0$.\nThe third question seems to remain open.\nReferences\n\n\n[Be91] Beck, J., The modulus of polynomials with zeros on the unit circle: A problem of Erdos. Annals of Math. (1991), 609-651.\n\n[Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155--180.\n\n[Li77] Linden, C. N., The modulus of polynomials with zeros on the unit circle. Bull. London Math. Soc. (1977), 65--69.\n\n[Wa80] Wagner, Gerold, On a problem of {E}rd\\H{o}s in {D}iophantine approximation. Bull. London Math. Soc. (1980), 81--88.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1940, "problem_number": "EP-120", "title": "Erdős Problem #120", "statement": "Let $A\\subseteq\\mathbb{R}$ be an infinite set. Must there be a set $E\\subset \\mathbb{R}$ of positive measure which does not contain any set of the shape $aA+b$ for some $a,b\\in\\mathbb{R}$ and $a\neq 0$?", "background": "The Erdos similarity problem.\nThis is true if $A$ is unbounded or dense in some interval. It therefore suffices to prove this when $A=\\{a_1>a_2>\\cdots\\}$ is a countable strictly monotone sequence which converges to $0$.\nSteinhaus \\cite{St20} has proved this is false whenever $A$ is a finite set.\nThis conjecture is known in many special cases (but, for example, it is open when $A=\\{1,1/2,1/4,\\ldots\\}$, which is Problem 94 on Green's open problems list). For an overview of progress we recommend a nice survey by Svetic \\cite{Sv00} on this problem. A survey of more recent progress was written by Jung, Lai, and Mooroogen \\cite{JLM24}.\nReferences\n\n\n[JLM24] Y. Jung and C.-K. Lai and Y. Mooroogen, Some recent progress on the Erdos similarity conjecture. arXiv:2412.11062 (2024).\n\n[St20] Steinhaus, Hugo, Sur les distances des points dans les ensembles de measure positive. Fund. Math. (1920), 93-104.\n\n[Sv00] Svetic, R. E., The Erdos similarity problem: a survey. Real Anal. Exchange (2000/01), 525-539.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1941, "problem_number": "EP-122", "title": "Erdős Problem #122", "statement": "For which number theoretic functions $f$ is it true that, for any $F(n)$ such that $f(n)/F(n)\\to 0$ for almost all $n$, there are infinitely many $x$ such that $ \\frac{\\#\\{ n\\in \\mathbb{N} : n+f(n)\\in (x,x+F(x))\\}}{F(x)}\\to \\infty? $ ", "background": "Asked by Erdos, Pomerance, and S\\'{a}rk\"{o}zy \\cite{EPS97} who prove that this is true when $f$ is the divisor function or the number of distinct prime divisors of $n$, but Erdos believed it is false when $f(n)=\\phi(n)$ or $\\sigma(n)$.\nReferences\n\n\n[EPS97] Erdos, Paul and Pomerance, Carl and S\\'{a}rk\"{o}zy, Andr\\'{a}s, On locally repeated values of certain arithmetic functions. IV. Ramanujan J. (1997), 227-241.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1942, "problem_number": "EP-123", "title": "Erdős Problem #123", "statement": "Let $a,b,c\\geq 1$ be three integers which are pairwise coprime. Is every large integer the sum of distinct integers of the form $a^kb^lc^m$ ($k,l,m\\geq 0$), none of which divide any other?", "background": "A sequence is said to be $d$-complete if every large integer is the sum of distinct integers from the sequence, none of which divide any other. This particular case of $d$-completeness was conjectured by Erdos and Lewin \\cite{ErLe96}, who (among other related results) prove this when $a=3$, $b=5$, and $c=7$.\nAs a partial record of progress so far, the sequence $\\{a^kb^lc^m\\}$ is known to be $d$-complete when:\n{UL}\n{LI}$a=3$, $b=5$, $c=7$ (Erdos and Lewin \\cite{ErLe96}).{/LI}\n{LI}$a=2$, $b=5$, $c\\in \\{7,11,13,17,19\\}$ (Erdos and Lewin \\cite{ErLe96}).{/LI}\n{LI}$a=2$, $b=5$, $c\\in \\{9,21,23,27,29,31\\}$ - more generally, $a=2$, $b=5$, and any $c>6$ with $(c,10)=1$ such that there exists $N$ where every integer in $(N,25cN)$ is the sum of distinct elements of $\\{2^k3^lc^m\\}$, none of which divide any other (Ma and Chen \\cite{MaCh16}).{/LI}\n{LI} $a=2$, $b=5$, $3\\leq c\\leq 87$ with $(c,10)=1$, or $a=2$, $b=7$, $3\\leq c\\leq 33$ with $(c,14)=1$, or $a=3$, $b=5$, $2\\leq c\\leq 14$ with $(c,15)=1$ (Chen and Yu \\cite{ChYu23b}).{/LI}\n{/UL}\nIn \\cite{Er92b} Erdos makes the stronger conjecture (for $a=2$, $b=3$, and $c=5$) that, for any $\\epsilon>0$, all large integers $n$ can be written as the sum of distinct integers $b_1<\\cdots 0$, of an infinite set of $d_i$ for which every sufficiently large integer can be written as a finite sum of the shape $\\sum_i c_ia_i$ where $c_i\\in \\{0,1\\}$ and $a_i\\in P(d_i,0)$ and yet $\\sum_{i}\\frac{1}{d_i-1}<\\epsilon$.\nSee also [125].\nReferences\n\n\n[BEGL96] Burr, S. A. and Erdos, P. and Graham, R. L. and Li, W. Wen-Ching, Complete sequences of sets of integer powers. Acta Arith. (1996), 133-138.\n\n[Er97] Erdos, Paul, Problems in number theory. New Zealand J. Math. (1997), 155-160.\n\n[Er97e] Erdos, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537.\n\n[Me04] Melfi, Giuseppe, On certain positive integer sequences. Riv. Mat. Univ. Parma (7) (2004), 253--260.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1944, "problem_number": "EP-125", "title": "Erdős Problem #125", "statement": "Let $A = \\{ \\sum\\epsilon_k3^k : \\epsilon_k\\in \\{0,1\\}\\}$ be the set of integers which have only the digits $0,1$ when written base $3$, and $B=\\{ \\sum\\epsilon_k4^k : \\epsilon_k\\in \\{0,1\\}\\}$ be the set of integers which have only the digits $0,1$ when written base $4$.\nDoes $A+B$ have positive density?", "background": "A problem of Burr, Erdos, Graham, and Li \\cite{BEGL96}. More generally, if $n_1<\\cdots1 $ and $A_i$ is the set of integers with only the digits $0,1$ in base $n_i$ then does $A_1+\\cdots+A_k$ have positive density? Melfi \\cite{Me01} noted this is false as written, with a counterexample given by $\\{3,9,81\\}$, but suggests it is true if we further insist that the $n_k$ are pairwise coprime.\nIf $C=A+B$ then Melfi \\cite{Me01} showed $\\lvert C\\cap[1,x]\\rvert \\gg x^{0.965}$ and Hasler and Melfi \\cite{HaMe24} improved this to $\\lvert C\\cap [1,x]\\rvert \\gg x^{0.9777}$. Hasler and Melfi also show that the lower density of $C$ is at most $ \\frac{1015}{1458}\\approx 0.69616. $ See also [124].\nReferences\n\n\n[BEGL96] Burr, S. A. and Erdos, P. and Graham, R. L. and Li, W. Wen-Ching, Complete sequences of sets of integer powers. Acta Arith. (1996), 133-138.\n\n[HaMe24] M. Hasler and G. Melfi, On sums of distinct powers of $3$ and $4$. Combinatorics and Number Theory (2024).\n\n[Me01] Melfi, Giuseppe, An additive problem about powers of fixed integers. Rend. Circ. Mat. Palermo (2) (2001), 239--246.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1945, "problem_number": "EP-126", "title": "Erdős Problem #126", "statement": "Let $f(n)$ be maximal such that if $A\\subseteq\\mathbb{N}$ has $\\lvert A\\rvert=n$ then $\\prod_{a\neq b\\in A}(a+b)$ has at least $f(n)$ distinct prime factors. Is it true that $f(n)/\\log n\\to\\infty$?", "background": "Investigated by Erdos and Tur\\'{a}n \\cite{ErTu34} (prompted by a question of L\\'{a}z\\'{a}r and Gr\"{u}nwald) in their first joint paper, where they proved that $ \\log n \\ll f(n) \\ll n/\\log n $ (the upper bound is trivial, taking $A=\\{1,\\ldots,n\\}$). Erdos says that $f(n)=o(n/\\log n)$ has never been proved, but perhaps never seriously attacked.\nThis problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.\nReferences\n\n\n[ErTu34] Erdos, Paul and Turan, Paul, On a Problem in the Elementary Theory of Numbers. Amer. Math. Monthly (1934), 608-611.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1946, "problem_number": "EP-129", "title": "Erdős Problem #129", "statement": "Let $R(n;k,r)$ be the smallest $N$ such that if the edges of $K_N$ are $r$-coloured then there is a set of $n$ vertices which does not contain a copy of $K_k$ in at least one of the $r$ colours. Prove that there is a constant $C=C(r)>1$ such that $ R(n;3,r) < C^{\\sqrt{n}}. $ ", "background": "Conjectured by Erdos and Gy\\'{a}rf\\'{a}s, who proved the existence of some $C>1$ such that $R(n;3,r)>C^{\\sqrt{n}}$. Note that when $r=k=2$ we recover the classic Ramsey numbers. Erdos thought it likely that for all $r,k\\geq 2$ there exists some $C_1,C_2>1$ (depending only on $r$) such that $ C_1^{n^{1/k-1}}< R(n;k,r) < C_2^{n^{1/k-1}}. $ Antonio Girao has pointed out that this problem as written is easily disproved, and indeed $R(n;3,2) \\geq C^{n}$:\nThe obvious probabilistic construction (randomly colour the edges red/blue independently uniformly at random) yields a 2-colouring of the edges of $K_N$ such every set on $n$ vertices contains a red triangle and a blue triangle (using that every set of $n$ vertices contains $\\gg n^2$ edge-disjoint triangles), provided $N \\leq C^n$ for some absolute constant $C>1$. This implies $R(n;3,2) \\geq C^{n}$, contradicting the conjecture.\nPerhaps Erdos had a different problem in mind, but it is not clear what that might be. It would presumably be one where the natural probabilistic argument would deliver a bound like $C^{\\sqrt{n}}$ as Erdos and Gy\\'{a}rf\\'{a}s claim to have achieved via the probabilistic method.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1947, "problem_number": "EP-130", "title": "Erdős Problem #130", "statement": "Let $A\\subset\\mathbb{R}^2$ be an infinite set which contains no three points on a line and no four points on a circle. Consider the graph with vertices the points in $A$, where two vertices are joined by an edge if and only if they are an integer distance apart.\nHow large can the chromatic number and clique number of this graph be? In particular, can the chromatic number be infinite?", "background": "Asked by Andr\\'{a}sfai and Erdos. Erdos \\cite{Er97b} also asked where such a graph could contain an infinite complete graph, but this is impossible by an earlier result of Anning and Erdos \\cite{AnEr45}.\nSee also [213].\nReferences\n\n\n[AnEr45] Anning, Norman H. and Erdos, Paul, Integral distances. Bull. Amer. Math. Soc. (1945), 598-600.\n\n[Er97b] Erdos, Paul, Some old and new problems in various branches of combinatorics. Discrete Math. (1997), 227-231.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1948, "problem_number": "EP-131", "title": "Erdős Problem #131", "statement": "Let $F(N)$ be the maximal size of $A\\subseteq\\{1,\\ldots,N\\}$ such that no $a\\in A$ divides the sum of any distinct elements of $A\\backslash\\{a\\}$. Estimate $F(N)$. In particular, is it true that $ F(N) > N^{1/2-o(1)}? $ ", "background": "This was studied by Erdos, Lev, Rauzy, S\\'{a}ndor, and S\\'{a}rk\"{o}zy \\cite{ELRSS99}, where they call such a property 'non-dividing', and prove the explicit bound $ F(N)<3N^{1/2}+1. $ In \\cite{Er97b} Erdos credits Csaba with a construction that proves $F(N) \\gg N^{1/5}$. Such a construction was also given in \\cite{ELRSS99}, where it is linked to the problem of non-averaging sets (see [186]).\nIndeed, every such set is non-averaging, and hence the result of Pham and Zakharov \\cite{PhZa24} implies $ F(N) \\leq N^{1/4+o(1)}. $ This shows the answer to the original question is no, but the general question of the correct growth of $F(N)$ remains open.\nIn \\cite{Er75b} Erdos writes that he originally thought $F(N) <(\\log N)^{O(1)}$, but that Straus proved that $ F(N) > \\exp((\\sqrt{\\tfrac{2}{\\log 2}}+o(1))\\sqrt{\\log N}). $ See also [13].\nThis is discussed in problem C16 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[ELRSS99] Erdos, P. and Lev, V. and Rauzy, G. and S\\'andor, C. and\nS\\'ark\"ozy, A., Greedy algorithm, arithmetic progressions, subset sums and\ndivisibility. Discrete Math. (1999), 119--135.\n\n[Er75b] Erdos, Paul, Problems and results in combinatorial number theory. Journ\\'{e}es Arithm\\'{e}tiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974) (1975), 295-310.\n\n[Er97b] Erdos, Paul, Some old and new problems in various branches of combinatorics. Discrete Math. (1997), 227-231.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[PhZa24] Pham, H. T. and Zakharov, D., Sharp bound for the Erdos-Straus non-averaging set problem. arXiv:2410.14624 (2024).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1949, "problem_number": "EP-132", "title": "Erdős Problem #132", "statement": "Let $A\\subset \\mathbb{R}^2$ be a set of $n$ points. Must there be two distances which occur at least once but between at most $n$ pairs of points? Must the number of such distances $\\to \\infty$ as $n\\to \\infty$?", "background": "Asked by Erdos and Pach. Hopf and Pannowitz \\cite{HoPa34} proved that the largest distance between points of $A$ can occur at most $n$ times, but it is unknown whether a second such distance must occur.\nIt may be true that there are at least $n^{1-o(1)}$ many such distances. In \\cite{Er97e} Erdos offers \\$100 for 'any nontrivial result'.\nErdos \\cite{Er84c} believed that for $n\\geq 5$ there must always exist at least two such distances. This is false for $n=4$, as witnessed by two equilateral triangles of the same side-length glued together. Erdos and Fishburn \\cite{ErFi95} proved this is true for $n=5$ and $n=6$.\nClemen, Dumitrescu, and Liu \\cite{CDL25} have proved that there always at least two such distances if $A$ is in convex position (that is, no point lies inside the convex hull of the others). They also prove it is true if the set $A$ is 'not too convex', in a specific technical sense.\nSee also [223], [756], and [957].\nReferences\n\n\n[CDL25] F. Clemen, A. Dumitrescu, and D. Liu, On multiplicities of interpoint distances. arXiv:2505.04283 (2025).\n\n[Er84c] Erdos, Paul, Some old and new problems in combinatorial geometry. Convexity and graph theory (Jerusalem, 1981) (1984), 129-136.\n\n[Er97e] Erdos, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537.\n\n[ErFi95] Erdos, Paul and Fishburn, Peter C., Multiplicities of interpoint distances in finite planar sets. Discrete Appl. Math. (1995), 141--147.\n\n[HoPa34] Hopf, H. and Pannwitz, E., Aufgabe 167. Jber. Deutsch. Math. Verein. (1934), 114.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1950, "problem_number": "EP-137", "title": "Erdős Problem #137", "statement": "We say that $N$ is powerful if whenever $p\\mid N$ we also have $p^2\\mid N$. Let $k\\geq 3$. Can the product of any $k$ consecutive positive integers ever be powerful?", "background": "Conjectured by Erdos and Selfridge. There are infinitely many $n$ such that $n(n+1)$ is powerful (see [364]). Erdos and Selfridge \\cite{ErSe75} proved that the product of $k\\geq 3$ consecutive positive integers can never be a perfect power. Erdos remarked that this 'seems hopeless at present'.\nIn \\cite{Er82c} he further conjectures that, if $k$ is fixed and $n$ is sufficiently large, then, for all $m$, there must be at least $k$ distinct primes $p$ such that $ p\\mid m(m+1)\\cdots (m+n) $ and yet $p^2$ does not divide the right-hand side.\nSee also [364].\nReferences\n\n\n[Er82c] Erdos, P., Miscellaneous problems in number theory. Congr. Numer. (1982), 25-45.\n\n[ErSe75] Erdos, P. and Selfridge, J. L., The product of consecutive integers is never a power. Illinois J. Math. (1975), 292-301.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1951, "problem_number": "EP-138", "title": "Erdős Problem #138", "statement": "Let the van der Waerden number $W(k)$ be such that whenever $N\\geq W(k)$ and $\\{1,\\ldots,N\\}$ is $2$-coloured there must exist a monochromatic $k$-term arithmetic progression. Improve the bounds for $W(k)$ - for example, prove that $W(k)^{1/k}\\to \\infty$.", "background": "When $p$ is prime Berlekamp \\cite{Be68} has proved $W(p+1)\\geq p2^p$. Gowers \\cite{Go01} has proved $ W(k) \\leq 2^{2^{2^{2^{2^{k+9}}}}}. $ The best general lower bound is $W(k)\\gg 2^k$, due to Kozik and Shabanov \\cite{KoSh16}.\nIn \\cite{Er81} Erdos further asks whether $W(k+1)/W(k)\\to \\infty$, or $W(k+1)-W(k)\\to \\infty$.\nIn \\cite{Er80} Erdos asks whether $W(k)/2^k\\to \\infty$, and offers \\$500 for a proof or disproof of $W(k)^{1/k}\\to \\infty$.\nReferences\n\n\n[Be68] Berlekamp, E. R., A construction for partitions which avoid long arithmetic progressions. Canad. Math. Bull. (1968), 409-414.\n\n[Er80] Erdos, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.\n\n[Er81] Erdos, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.\n\n[Go01] Gowers, W. T., A new proof of Szemer\\'{e}di's theorem. Geom. Funct. Anal. (2001), 465-588.\n\n[KoSh16] Kozik, Jakub and Shabanov, Dmitry, Improved algorithms for colorings of simple hypergraphs and\napplications. J. Combin. Theory Ser. B (2016), 312--332.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1952, "problem_number": "EP-141", "title": "Erdős Problem #141", "statement": "Let $k\\geq 3$. Are there $k$ consecutive primes in arithmetic progression?", "background": "Green and Tao \\cite{GrTa08} have proved that there must always exist some $k$ primes in arithmetic progression, but these need not be consecutive. Erdos called this conjecture 'completely hopeless at present'.\nThe existence of such progressions for small $k$ has been verified for $k\\leq 10$, see the Wikipedia page. It is open, even for $k=3$, whether there are infinitely many such progressions.\nSee also [219].\nThis is discussed in problem A6 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[GrTa08] Green, Ben and Tao, Terence, The primes contain arbitrarily long arithmetic progressions. Ann. of Math. (2) (2008), 481-547.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L3\"\n},{", "difficulty_level_id": 3, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1953, "problem_number": "EP-142", "title": "Erdős Problem #142", "statement": "Let $r_k(N)$ be the largest possible size of a subset of $\\{1,\\ldots,N\\}$ that does not contain any non-trivial $k$-term arithmetic progression. Prove an asymptotic formula for $r_k(N)$.", "background": "Erdos remarked this is 'probably unattackable at present'. In \\cite{Er97c} Erdos offered \\$1000, but given that he elsewhere offered \\$5000 just for (essentially) showing that $r_k(N)=o_k(N/\\log N)$, that value seems odd. In \\cite{Er81} he offers \\$10000, stating it is 'probably enormously difficult'.\nThe best known upper bounds for $r_k(N)$ are due to Kelley and Meka \\cite{KeMe23} for $k=3$, Green and Tao \\cite{GrTa17} for $k=4$, and Leng, Sah, and Sawhney \\cite{LSS24} for $k\\geq 5$. An asymptotic formula is still far out of reach, even for $k=3$.\nSee also [3] and [139].\nReferences\n\n\n[Er81] Erdos, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.\n\n[Er97c] Erdos, Paul, Some of my favorite problems and results. The mathematics of Paul Erdos, I (1997), 47-67.\n\n[GrTa17] Green, Ben and Tao, Terence, New bounds for Szemer\\'{e}di's theorem, III: a polylogarithmic bound for $r_4(N)$. Mathematika (2017), 944-1040.\n\n[KeMe23] Kelley, Z. and Meka, R., Strong Bounds for 3-Progressions. arXiv:2302.05537 (2023).\n\n[LSS24] Leng, J., Sah, A. and Sawhney, M., Improved bounds for Szemer\\'{e}di's theorem. arXiv:2402.17995 (2024).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1954, "problem_number": "EP-143", "title": "Erdős Problem #143", "statement": "Let $A\\subset (1,\\infty)$ be a countably infinite set such that for all $x\neq y\\in A$ and integers $k\\geq 1$ we have $ \\lvert kx -y\\rvert \\geq 1. $ Does this imply that $A$ is sparse? In particular, does this imply that $ \\sum_{x\\in A}\\frac{1}{x\\log x}<\\infty $ or $ \\sum_{\\substack{x 0$ is some absolute constant and $c_0=1.26408\\cdots$ is the 'Vardi constant'. The lower bound is due to Konyagin \\cite{Ko14} and the upper bound to Elsholtz and Planitzer \\cite{ElPl21}.\nReferences\n\n\n[ElPl21] Elsholtz, Christian and Planitzer, Stefan, Sums of four and more unit fractions and approximate parametrizations. Bull. Lond. Math. Soc. (2021), 695-709.\n\n[Ko14] Konyagin, S. V., Double exponential lower bound for the number of representations of unity by Egyptian fractions. Math. Notes (2014), 277-281.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1958, "problem_number": "EP-149", "title": "Erdős Problem #149", "statement": "Let $G$ be a graph with maximum degree $\\Delta$. Is $G$ the union of at most $\\tfrac{5}{4}\\Delta^2$ sets of strongly independent edges (sets such that the induced subgraph is the union of vertex-disjoint edges)?", "background": "Asked by Erdos and Ne\\v{s}et\\v{r}il in 1985 (see \\cite{FGST89}). This is equivalent to asking whether the chromatic number of the square of the line graph $L(G)^2$ is at most $\\frac{5}{4}\\Delta^2$.\nThis bound would be the best possible, as witnessed by a blowup of $C_5$. The minimum number of such sets required is sometimes called the strong chromatic index of $G$.\nThe weaker conjecture that there exists some $c>0$ such that $(2-c)\\Delta^2$ sets suffice was proved by Molloy and Reed \\cite{MoRe97}, who proved that $1.998\\Delta^2$ sets suffice (for $\\Delta$ sufficiently large). This was improved to $1.93\\Delta^2$ by Bruhn and Joos \\cite{BrJo18} and to $1.835\\Delta^2$ by Bonamy, Perrett, and Postle \\cite{BPP22}. The best bound currently available is $ 1.772\\Delta^2, $ proved by Hurley, de Joannis de Verclos, and Kang \\cite{HJK22}. Mahdian has, in their Masters' thesis, proved an upper bound of $(2+o(1))\\frac{\\Delta^2}{\\log \\Delta}$ under the additional assumption that $G$ is $C_4$-free.\nErdos and Ne\\v{s}et\\v{r}il also asked the easier problem of whether $G$ containing at least $\\tfrac{5}{4}\\Delta^2$ many edges implies $G$ containing two strongly independent edges. This was proved by Chung, Gy\\'{a}rf\\'{a}s, Tuza, and Trotter \\cite{CGTT90}.\nIt is still open even whether the clique number of $L(G)^2$ at most $\\frac{5}{4}\\Delta^2$. Let $\\omega=\\omega(L(G)^2)$ be this clique number. \\'{S}leszy\\'{n}ska-Nowak \\cite{Sl15} proved $\\omega \\leq \\frac{3}{2}\\Delta^2$. Faron and Postle \\cite{FaPo19} proved $\\omega\\leq \\frac{4}{3}\\Delta^2$. Cames van Batenburg, Kang, and Pirot \\cite{CKP20} have proved $\\omega\\leq \\frac{5}{4}\\Delta^2$ under the additional assumption that $G$ is triangle-free (and $\\omega\\leq \\Delta^2$ if $G$ is $C_5$-free).\nReferences\n\n\n[BPP22] Bonamy, Marthe and Perrett, Thomas and Postle, Luke, Colouring graphs with sparse neighbourhoods: bounds and\napplications. J. Combin. Theory Ser. B (2022), 278-317.\n\n[BrJo18] Bruhn, Henning and Joos, Felix, A stronger bound for the strong chromatic index. Combin. Probab. Comput. (2018), 21-43.\n\n[CGTT90] Chung, F. R. K. and Gy\\'arf\\'as, A. and Tuza, Z. and Trotter,\nW. T., The maximum number of edges in {$2K_2$}-free graphs of bounded\ndegree. Discrete Math. (1990), 129--135.\n\n[CKP20] Cames van Batenburg, Wouter and Kang, Ross J. and Pirot,\nFran\\c cois, Strong cliques and forbidden cycles. Indag. Math. (N.S.) (2020), 64--82.\n\n[FGST89] Faudree, R. J. and Gy\\'{a}rf\\'{a}s, A. and Schelp, R. H. and Tuza,\nZs., Induced matchings in bipartite graphs. Discrete Math. (1989), 83-87.\n\n[FaPo19] Faron, Maxime and Postle, Luke, On the clique number of the square of a line graph and its\nrelation to maximum degree of the line graph. J. Graph Theory (2019), 261--274.\n\n[HJK22] Hurley, Eoin and de Joannis de Verclos, R\\'{e}mi and Kang, Ross\nJ., An improved procedure for colouring graphs of bounded local\ndensity. Adv. Comb. (2022), Paper No. 7, 33.\n\n[MoRe97] Molloy, Michael and Reed, Bruce, A bound on the strong chromatic index of a graph. J. Combin. Theory Ser. B (1997), 103-109.\n\n[Sl15] No reference found.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1959, "problem_number": "EP-151", "title": "Erdős Problem #151", "statement": "For a graph $G$ let $\\tau(G)$ denote the minimal number of vertices that include at least one from each maximal clique of $G$ on at least two vertices (sometimes called the clique transversal number).\nLet $H(n)$ be maximal such that every triangle-free graph on $n$ vertices contains an independent set on $H(n)$ vertices.\nIf $G$ is a graph on $n$ vertices then is $ \\tau(G)\\leq n-H(n)? $ ", "background": "It is easy to see that $\\tau(G) \\leq n-\\sqrt{n}$. Note also that if $G$ is triangle-free then trivially $\\tau(G)\\leq n-H(n)$.\nThis is listed in \\cite{Er88} as a problem of Erdos and Gallai, who were unable to make progress even assuming $G$ is $K_4$-free. There Erdos remarked that this conjecture is 'perhaps completely wrongheaded'.\nIt later appeared as Problem 1 in \\cite{EGT92}.\nThe general behaviour of $\\tau(G)$ is the subject of [610].\nReferences\n\n\n[EGT92] Erdos, Paul and Gallai, Tibor and Tuza, Zsolt, Covering the cliques of a graph with vertices. Discrete Math. (1992), 279-289.\n\n[Er88] Erdos, P, Problems and results in combinatorial analysis and graph theory. Discrete Math. (1988), 81-92.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1960, "problem_number": "EP-152", "title": "Erdős Problem #152", "statement": "For any $M\\geq 1$, if $A\\subset \\mathbb{N}$ is a sufficiently large finite Sidon set then there are at least $M$ many $a\\in A+A$ such that $a+1,a-1\not\\in A+A$.", "background": "There may even be $\\gg \\lvert A\\rvert^2$ many such $a$. A similar question can be asked for truncations of infinite Sidon sets.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1961, "problem_number": "EP-153", "title": "Erdős Problem #153", "statement": "Let $A$ be a finite Sidon set and $A+A=\\{s_1<\\cdots0$ such that\n$$R(C_4,K_n) \\ll n^{2-c}.$$", "background": "The current bounds are $ \\frac{n^{3/2}}{(\\log n)^{3/2}}\\ll R(C_4,K_n)\\ll \\frac{n^2}{(\\log n)^2}. $ The upper bound is due to Szemer\\'{e}di (mentioned in \\cite{EFRS78}), and the lower bound is due to Spencer \\cite{Sp77}.\nThis problem is #17 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[EFRS78] Erdos, Paul and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., On cycle-complete graph Ramsey numbers. J. Graph Theory (1978), 53-64.\n\n[Sp77] Spencer, J., Asymptotic lower bounds for Ramsey functions. Discrete Math. (1977), 69-76.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1966, "problem_number": "EP-160", "title": "Erdős Problem #160", "statement": "Let $h(N)$ be the smallest $k$ such that $\\{1,\\ldots,N\\}$ can be coloured with $k$ colours so that every four-term arithmetic progression must contain at least three distinct colours. Estimate $h(N)$.", "background": "Investigated by Erdos and Freud. This has been discussed on MathOverflow, where LeechLattice shows $ h(N) \\ll N^{2/3}. $ In the comments of this site Hunter improves this to $ h(N) \\ll N^{\\frac{\\log 3}{\\log 22}+o(1)} $ (note $\\frac{\\log 3}{\\log 22}\\approx 0.355$).\nThe observation of Zach Hunter in that question coupled with recent progress on the size of subsets without three-term arithmetic progression (see \\cite{BlSi23} which improves slightly on the bounds due to Kelley and Meka \\cite{KeMe23}) imply that $ h(N) \\gg \\exp(c(\\log N)^{1/9}) $ for some $c>0$.\nReferences\n\n\n[BlSi23] T. F. Bloom and O. Sisask, An improvement to the Kelley-Meka bounds on three-term arithmetic progressions. arXiv:2309.02353 (2023).\n\n[KeMe23] Kelley, Z. and Meka, R., Strong Bounds for 3-Progressions. arXiv:2302.05537 (2023).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1967, "problem_number": "EP-161", "title": "Erdős Problem #161", "statement": "Let $\\alpha\\in[0,1/2)$ and $n,t\\geq 1$. Let $F^{(t)}(n,\\alpha)$ be the smallest $m$ such that we can $2$-colour the edges of the complete $t$-uniform hypergraph on $n$ vertices such that if $X\\subseteq [n]$ with $\\lvert X\\rvert \\geq m$ then there are at least $\\alpha \\binom{\\lvert X\\rvert}{t}$ many $t$-subsets of $X$ of each colour.\nFor fixed $n,t$ as we change $\\alpha$ from $0$ to $1/2$ does $F^{(t)}(n,\\alpha)$ increase continuously or are there jumps? Only one jump?", "background": "For $\\alpha=0$ this is the usual Ramsey function.\nA conjecture of Erdos, Hajnal, and Rado (see [562]) implies that $ F^{(t)}(n,0)\\asymp \\log_{t-1} n $ and results of Erdos and Spencer imply that $ F^{(t)}(n,\\alpha) \\gg_\\alpha (\\log n)^{\\frac{1}{t-1}} $ for all $\\alpha>0$, and a similar upper bound holds for $\\alpha$ close to $1/2$.\nErdos said in \\cite{Er90b}: 'If I can hazard a guess completely unsupported by evidence, I am afraid that the jump occurs all in one step at $0$. It would be much more interesting if my conjecture would be wrong and perhaps there is some hope for this for $t>3$. I know nothing and offer \\$500 to anybody who can clear up this mystery.'\nConlon, Fox, and Sudakov \\cite{CFS11} have proved that, for any fixed $\\alpha>0$, $ F^{(3)}(n,\\alpha) \\ll_\\alpha \\sqrt{\\log n}. $ Coupled with the lower bound above, this implies that there is only one jump for fixed $\\alpha$ when $t=3$, at $\\alpha=0$.\nFor all $\\alpha>0$ it is known that $ F^{(t)}(n,\\alpha)\\gg_t (\\log n)^{c_\\alpha}. $ See also [563] for more on the case $t=2$.\nThis problem is #40 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[CFS11] Conlon, David and Fox, Jacob and Sudakov, Benny, Large almost monochromatic subsets in hypergraphs. Israel J. Math. (2011), 423--432.\n\n[Er90b] Erdos, Paul, Problems and results on graphs and hypergraphs: similarities and differences. Mathematics of Ramsey theory (1990), 12-28.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1968, "problem_number": "EP-162", "title": "Erdős Problem #162", "statement": "Let $\\alpha>0$ and $n\\geq 1$. Let $F(n,\\alpha)$ be the largest $k$ such that there exists some 2-colouring of the edges of $K_n$ in which any induced subgraph $H$ on at least $k$ vertices contains more than $\\alpha\\binom{\\lvert H\\rvert}{2}$ many edges of each colour.\nProve that for every fixed $0\\leq \\alpha \\leq 1/2$, as $n\\to\\infty$, $ F(n,\\alpha)\\sim c_\\alpha \\log n $ for some constant $c_\\alpha$.", "background": "It is easy to show with the probabilistic method that there exist $c_1(\\alpha),c_2(\\alpha)$ such that $ c_1(\\alpha)\\log n < F(n,\\alpha) < c_2(\\alpha)\\log n. $ \",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1969, "problem_number": "EP-165", "title": "Erdős Problem #165", "statement": "Give an asymptotic formula for $R(3,k)$.", "background": "It is known that there exists some constant $c>0$ such that for large $k$ $ (c+o(1))\\frac{k^2}{\\log k}\\leq R(3,k) \\leq (1+o(1))\\frac{k^2}{\\log k}. $ The lower bound is due to Kim \\cite{Ki95}, the upper bound is due to Shearer \\cite{Sh83}, improving an earlier bound of Ajtai, Koml\\'{o}s, and Szemer\\'{e}di \\cite{AKS80}.\nThe value of $c$ in the lower bound has seen a number of improvements. Kim's original proof gave $c\\geq 1/162$. The bound $c\\geq 1/4$ was proved independently by Bohman and Keevash \\cite{BoKe21} and Pontiveros, Griffiths and Morris \\cite{PGM20}. The latter collection of authors conjecture that this lower bound is the true order of magnitude.\nThis was, however, improved by Campos, Jenssen, Michelen, and Sahasrabudhe \\cite{CJMS25} to $c\\geq 1/3$, and further by Hefty, Horn, King, and Pfender \\cite{HHKP25} to $c\\geq 1/2$. Both of these papers conjecture that $c=1/2$ is the correct asymptotic.\nSee also [544], and [986] for the general case. See [1013] for a related function.\nReferences\n\n\n[AKS80] Ajtai, Mikl\\'{o}s and Koml\\'{o}s, J\\'{a}nos and Szemer\\'{e}di, Endre, A note on Ramsey numbers. J. Combin. Theory Ser. A (1980), 354-360.\n\n[BoKe21] Bohman, Tom and Keevash, Peter, Dynamic concentration of the triangle-free process. Random Structures Algorithms (2021), 221-293.\n\n[CJMS25] M. Campos, M. Jenssen, M. Michelen, and J. Sahasrabudhe, A new lower bound for the Ramsey numbers $R(3,k)$. arXiv:2505.13371 (2025).\n\n[HHKP25] Z. Hefty, P. Horn, D. King, and F. Pfender, Improving $R(3,k)$ in just two bites. arXiv:2510.19718 (2025).\n\n[Ki95] Kim, J. H., The Ramsey number $R(3,t)$ has order of magnitude $t^2/\\log t$. Random Structures and Algorithms (1995), 173-207.\n\n[PGM20] Fiz Pontiveros, Gonzalo and Griffiths, Simon and Morris, Robert, The triangle-free process and the Ramsey number $R(3,k)$. Mem. Amer. Math. Soc. (2020), v+125.\n\n[Sh83] Shearer J., A note on the independence number of triangle-free graphs. Discrete Math. (1983), 83-87.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1970, "problem_number": "EP-168", "title": "Erdős Problem #168", "statement": "Let $F(N)$ be the size of the largest subset of $\\{1,\\ldots,N\\}$ which does not contain any set of the form $\\{n,2n,3n\\}$. What is $ \\lim_{N\\to \\infty}\\frac{F(N)}{N}? $ Is this limit irrational?", "background": "This limit was proved to exist by Graham, Spencer, and Witsenhausen \\cite{GSW77}, who showed it is equal to $ \\frac{1}{3}\\sum_{k\\in K}\\frac{1}{d_k}, $ where $d_1f(k-1)$, where $f$ counts the largest subset of $\\{d_1,\\ldots,d_k\\}$ that avoids $\\{n,2n,3n\\}$.\nSimilar questions can be asked for the density or upper density of infinite sets without such configurations.\nThe limit can be estimated by elementary arguments (see the comments). Eberhard has used the formula of \\cite{GSW77} mentioned above to calculate the value of the limit as $ 0.800965\\cdots. $ This problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.\nReferences\n\n\n[GSW77] Graham, R. and Spencer, J. and Witsenhausen, H., On Extremal Density Theorems for Linear Forms. Number Theory and Algebra (1977).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1971, "problem_number": "EP-169", "title": "Erdős Problem #169", "statement": "Let $k\\geq 3$ and $f(k)$ be the supremum of $\\sum_{n\\in A}\\frac{1}{n}$ as $A$ ranges over all sets of positive integers which do not contain a $k$-term arithmetic progression. Estimate $f(k)$.\nIs $ \\lim_{k\\to \\infty}\\frac{f(k)}{\\log W(k)}=\\infty $ where $W(k)$ is the van der Waerden number?", "background": "Berlekamp \\cite{Be68} proved $f(k) \\geq \\frac{\\log 2}{2}k$. Gerver \\cite{Ge77} proved $ f(k) \\geq (1-o(1))k\\log k. $ It is trivial that $ \\frac{f(k)}{\\log W(k)}\\geq \\frac{1}{2}, $ but improving the right-hand side to any constant $>1/2$ is open.\nGerver also proved (see the comments for an alternative argument of Tao) that [3] is equivalent to $f(k)$ being finite for all $k$.\nThe current record for $f(3)$ is $f(3)\\geq 3.00849$, due to Wr\\'{o}blewski \\cite{Wr84}. Walker \\cite{Wa25} proved $f(4)\\geq 4.43975$.\nWalker \\cite{Wa25} has shown that it suffices to consider Kempner sets (that is, sets of integers defined as all those whose base $b$ digits are contained in some $S\\subset \\{0,\\ldots,b-1\\}$ for fixed $b$ and $S$), in the sense that for any $k\\geq 3$ and $\\epsilon>0$ there is a Kempner set $A$ lacking $k$-term arithmetic progressions such that $ \\sum_{n\\in A}\\frac{1}{n}\\geq f(k)-\\epsilon. $ \nReferences\n\n\n[Be68] Berlekamp, E. R., A construction for partitions which avoid long arithmetic progressions. Canad. Math. Bull. (1968), 409-414.\n\n[Ge77] Gerver, Joseph L., The sum of the reciprocals of a set of integers with no\narithmetic progression of {$k$} terms. Proc. Amer. Math. Soc. (1977), 211--214.\n\n[Wa25] A. Walker, Integer sets of large harmonic sum which avoid long arithmetic progressions. arXiv:2203.06045 (2025).\n\n[Wr84] No reference found.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1972, "problem_number": "EP-170", "title": "Erdős Problem #170", "statement": "Let $F(N)$ be the smallest possible size of $A\\subset \\{0,1,\\ldots,N\\}$ such that $\\{0,1,\\ldots,N\\}\\subset A-A$. Find the value of $ \\lim_{N\\to \\infty}\\frac{F(N)}{N^{1/2}}. $ ", "background": "The Sparse Ruler problem. R\\'{e}dei asked whether this limit exists, which was proved by Erdos and G\\'{a}l \\cite{ErGa48}. Bounds on the limit were improved by Leech \\cite{Le56}. The limit is known to be in the interval $[1.56,\\sqrt{3}]$. The lower bound is due to Leech \\cite{Le56}, the upper bound is due to Wichmann \\cite{Wi63}. Computational evidence by Pegg \\cite{Pe20} suggests that the upper bound is the truth. A similar question can be asked without the restriction $A\\subset \\{0,1,\\ldots,N\\}$.\nReferences\n\n\n[ErGa48] Erdos, P. and G\\'{a}l, I., On the representation of $1,2,\\ldots,N$ by differences. Nederl. Akad. Wetensch., Proc. (1948), 1155-1158.\n\n[Le56] Leech, J., On the representation of $1,2,\\ldots,n$ by differences. J. London Math. Soc. (1956), 160-169.\n\n[Pe20] Pegg, E., Hitting All the Marks: Exploring New Bounds for Sparse Rulers and a Wolfram Language Proof. https://blog.wolfram.com/2020/02/12/hitting-all-the-marks-exploring-new-bounds-for-sparse-rulers-and-a-wolfram-language-proof/ (2020).\n\n[Wi63] Wichmann, B., A note on restricted difference bases. J. London Math. Soc. (1963), 465-466.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1973, "problem_number": "EP-172", "title": "Erdős Problem #172", "statement": "Is it true that in any finite colouring of $\\mathbb{N}$ there exist arbitrarily large finite $A$ such that all sums and products of distinct elements in $A$ are the same colour?", "background": "First asked by Hindman. Hindman \\cite{Hi80} has proved this is false (with 7 colours) if we ask for an infinite $A$. In \\cite{Er77c} Erdos asks about the case for an infinite $A$ with just $2$ colours.\nMoreira \\cite{Mo17} has proved that in any finite colouring of $\\mathbb{N}$ there exist $x,y$ such that $\\{x,x+y,xy\\}$ are all the same colour.\nAlweiss \\cite{Al23} has proved that, in any finite colouring of $\\mathbb{Q}\\backslash \\{0\\}$ there exist arbitrarily large finite $A$ such that all sums and products of distinct elements in $A$ are the same colour. Bowen and Sabok \\cite{BoSa22} had proved this earlier for the first non-trivial case of $\\lvert A\\rvert=2$.\nReferences\n\n\n[Al23] R. Alweiss, Hindman's conjecture over the rationals. arXiv:2307.08901 (2023).\n\n[BoSa22] M. Bowen and M. Sabok, Monochromatic Sums and Products in the Rationals. arXiv:2210.12290 (2022).\n\n[Er77c] Erdos, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ.,\nNew York, 1976) (1977), 43-72.\n\n[Hi80] Hindman, Neil, Partitions and sums and products-two counterexamples. J. Combin. Theory Ser. A (1980), 113-120.\n\n[Mo17] Moreira, J., Monochromatic sums and products in $\\mathbbN$. Ann. Math. (2017), 1069-1090.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1974, "problem_number": "EP-173", "title": "Erdős Problem #173", "statement": "In any $2$-colouring of $\\mathbb{R}^2$, for all but at most one triangle $T$, there is a monochromatic congruent copy of $T$.", "background": "For some colourings a single equilateral triangle has to be excluded, considering the colouring by alternating strips. Shader \\cite{Sh76} has proved this is true if we just consider a single right-angled triangle.\nReferences\n\n\n[Sh76] Shader, L., All right triangles are Ramsey in $\\mathbbE^2$!. J. Comb. Th. A (1976), 385-389.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1975, "problem_number": "EP-174", "title": "Erdős Problem #174", "statement": "A finite set $A\\subset \\mathbb{R}^n$ is called Ramsey if, for any $k\\geq 1$, there exists some $d=d(A,k)$ such that in any $k$-colouring of $\\mathbb{R}^d$ there exists a monochromatic copy of $A$. Characterise the Ramsey sets in $\\mathbb{R}^n$.", "background": "Erdos, Graham, Montgomery, Rothschild, Spencer, and Straus \\cite{EGMRSS73} proved that every Ramsey set is 'spherical': it lies on the surface of some sphere. Graham has conjectured that every spherical set is Ramsey. Leader, Russell, and Walters \\cite{LRW12} have alternatively conjectured that a set is Ramsey if and only if it is 'subtransitive': it can be embedded in some higher-dimensional set on which rotations act transitively.\nSets known to be Ramsey include vertices of $k$-dimensional rectangles \\cite{EGMRSS73}, non-degenerate simplices \\cite{FrRo90}, trapezoids \\cite{Kr92}, and regular polygons/polyhedra \\cite{Kr91}.\nReferences\n\n\n[EGMRSS73] Erdos, P. and Graham, R. L. and Montgomery, P. and Rothschild, B. L. and Spencer, J. and Straus, E. G., Euclidean Ramsey Theorems I. J. Comb. Th. A (1973), 341-363.\n\n[FrRo90] Frankl, P. and R\"{o}dl, V., A partition property of simplices in Euclidean space. J. Amer. Math. Soc. (1990), 1-7.\n\n[Kr91] K\\v{r}\\'{\\i}\\v{z}, Igor, Permutation groups in Euclidean Ramsey theory. Proc. Amer. Math. Soc. (1991), 899-907.\n\n[Kr92] K\\v{r}\\'{\\i}\\v{z}, Igor, All trapezoids are Ramsey. Discrete Math. (1992), 59-62.\n\n[LRW12] Leader, Imre and Russell, Paul A. and Walters, Mark, Transitive sets in Euclidean Ramsey theory. J. Combin. Theory Ser. A (2012), 382-396.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1976, "problem_number": "EP-176", "title": "Erdős Problem #176", "statement": "Let $N(k,\\ell)$ be the minimal $N$ such that for any $f:\\{1,\\ldots,N\\}\\to\\{-1,1\\}$ there must exist a $k$-term arithmetic progression $P$ such that $ \\left\\lvert \\sum_{n\\in P}f(n)\\right\\rvert\\geq \\ell. $ Find good upper bounds for $N(k,\\ell)$. Is it true that for any $c>0$ there exists some $C>1$ such that $ N(k,ck)\\leq C^k? $ What about $ N(k,2)\\leq C^k $ or $ N(k,\\sqrt{k})\\leq C^k? $ ", "background": "When $\\ell=k$ this is the van der Waerden number $W(k)$ (see [138]). Spencer \\cite{Sp73} has proved that if $k=2^tm$ with $m$ odd then $ N(k,1)=2^t(k-1)+1. $ Erdos and Graham write that 'no decent bound' is known even for $N(k,2)$.\nErdos \\cite{Er63d} proved that, for every $c>0$, $ N(k,ck)> (1+\\alpha_c)^k $ where $\\alpha_c\\to 0$ as $c\\to 0$ and $\\alpha_c\\to \\sqrt{2}-1$ as $c\\to 1$.\nReferences\n\n\n[Er63d] Erdos, P\\'al, On combinatorial questions connected with a theorem of\n{R}amsey and van der {W}aerden. Mat. Lapok (1963), 29--37.\n\n[Sp73] J. Spencer, Problems 185. Bull. Canad. Math. Soc. (1973), 185.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1977, "problem_number": "EP-177", "title": "Erdős Problem #177", "statement": "Find the smallest $h(d)$ such that the following holds. There exists a function $f:\\mathbb{N}\\to\\{-1,1\\}$ such that, for every $d\\geq 1$, $ \\max_{P_d}\\left\\lvert \\sum_{n\\in P_d}f(n)\\right\\rvert\\leq h(d), $ where $P_d$ ranges over all finite arithmetic progressions with common difference $d$.", "background": "Cantor, Erdos, Schreiber, and Straus \\cite{Er66} proved that $h(d)\\ll d!$ is possible. Van der Waerden's theorem implies that $h(d)\\to \\infty$. Beck \\cite{Be17} has shown that $h(d) \\leq d^{8+\\epsilon}$ is possible for every $\\epsilon>0$. Roth's famous discrepancy lower bound \\cite{Ro64} implies that $h(d)\\gg d^{1/2}$.\nReferences\n\n\n[Be17] Beck, J\\'{o}zsef, A discrepancy problem: balancing infinite dimensional vectors. Number theory-Diophantine problems, uniform distribution\nand applications (2017), 61-82.\n\n[Er66] Erdos, P\\'al, Remarks on number theory. {V}. {E}xtremal problems in number\ntheory. {II}. Mat. Lapok (1966), 135--155.\n\n[Ro64] Roth, K. F., Remark concerning integer sequences. Acta Arith. (1964), 257-260.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1978, "problem_number": "EP-180", "title": "Erdős Problem #180", "statement": "If $\\mathcal{F}$ is a finite set of finite graphs then $\\mathrm{ex}(n;\\mathcal{F})$ is the maximum number of edges a graph on $n$ vertices can have without containing any subgraphs from $\\mathcal{F}$. Note that it is trivial that $\\mathrm{ex}(n;\\mathcal{F})\\leq \\mathrm{ex}(n;G)$ for every $G\\in\\mathcal{F}$.\nIs it true that, for every $\\mathcal{F}$, there exists $G\\in\\mathcal{F}$ such that $ \\mathrm{ex}(n;G)\\ll_{\\mathcal{F}}\\mathrm{ex}(n;\\mathcal{F})? $ ", "background": "A problem of Erdos and Simonovits.\nThis is trivially true if $\\mathcal{F}$ does not contain any bipartite graphs, since by the Erdos-Stone theorem if $H\\in\\mathcal{F}$ has minimal chromatic number $r\\geq 2$ then $ \\mathrm{ex}(n;H)=\\mathrm{ex}(n;\\mathcal{F})=\\left(\\frac{r-2}{r-1}+o(1)\\right)\\binom{n}{2}. $ Erdos and Simonovits observe that this is false for infinite families $\\mathcal{F}$, e.g. the family of all cycles.\nHunter has provided the following 'folklore counterexample': if $\\mathcal{F}=\\{H_1,H_2\\}$ where $H_1$ is a star and $H_2$ is a matching, both with at least two edges, then $\\mathrm{ex}(n;\\mathcal{F})\\ll 1$, but $\\mathrm{ex}(n;H_i)\\asymp n$ for $1\\leq i\\leq 2$. This conjecture may still hold for all other $\\mathcal{F}$.\nSee also [575].\nThis problem is #47 in Extremal Graph Theory in the graphs problem collection.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1979, "problem_number": "EP-181", "title": "Erdős Problem #181", "statement": "Let $Q_n$ be the $n$-dimensional hypercube graph (so that $Q_n$ has $2^n$ vertices and $n2^{n-1}$ edges). Prove that $ R(Q_n) \\ll 2^n. $ ", "background": "Conjectured by Burr and Erdos, althouhg in \\cite{Er93} Erdos says the behaviour of $R(Q_n)$ was considered by himself and S\\'{o}s, who could not decide whether $R(Q_n)/2^n\\to \\infty$ or not.\nThe trivial bound is $ R(Q_n) \\leq R(K_{2^n})\\leq C^{2^n} $ for some constant $C>1$. This was improved a number of times; the current best bound due to Tikhomirov \\cite{Ti22} is $ R(Q_n)\\ll 2^{(2-c)n} $ for some small constant $c>0$. (In fact $c\\approx 0.03656$ is permissible.)\nThis problem is #20 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[Er93] Erdos, Paul, Some of my favorite solved and unsolved problems in graph\ntheory. Quaestiones Math. (1993), 333-350.\n\n[Ti22] Tikhomirov, K., A remark on the Ramsey number of the hypercube. arXiv:2208.14568 (2022).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1980, "problem_number": "EP-183", "title": "Erdős Problem #183", "statement": "Let $R(3;k)$ be the minimal $n$ such that if the edges of $K_n$ are coloured with $k$ colours then there must exist a monochromatic triangle. Determine $ \\lim_{k\\to \\infty}R(3;k)^{1/k}. $ ", "background": "Erdos offers \\$100 for showing that this limit is finite. An easy pigeonhole argument shows that $ R(3;k)\\leq 2+k(R(3;k-1)-1), $ from which $R(3;k)\\leq \\lceil e k!\\rceil$ immediately follows. The best-known upper bounds are all of the form $ck!+O(1)$, and arise from this type of inductive relationship and computational bounds for $R(3;k)$ for small $k$. The best-known lower bound (coming from lower bounds for Schur numbers) is $ R(3,k)\\geq (380)^{k/5}-O(1), $ due to Ageron, Casteras, Pellerin, Portella, Rimmel, and Tomasik \\cite{ACPPRT21} (improving previous bounds of Exoo \\cite{Ex94} and Fredricksen and Sweet \\cite{FrSw00}). Note that $380^{1/5}\\approx 3.2806$.\nSee also [483].\nThis problem is #21 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[ACPPRT21] R. Ageron, P. Casteras, T. Pellerin, Y. Portella, A. Rimmel, and J. Tomasik, New lower bounds for Schur and weak Schur numbers. arXiv:2112.03175 (2021).\n\n[Ex94] Exoo, G., A lower bound for Schur numbers and multicolor Ramsey numbers. Electronic J. of Combinatorics (1994).\n\n[FrSw00] Fredricksen, Harold and Sweet, Melvin M., Symmetric sum-free partitions and lower bounds for {S}chur\nnumbers. Electron. J. Combin. (2000), Research Paper 32, 9.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1981, "problem_number": "EP-184", "title": "Erdős Problem #184", "statement": "Any graph on $n$ vertices can be decomposed into $O(n)$ many edge-disjoint cycles and edges.", "background": "Conjectured by Erdos and Gallai, who proved that $O(n\\log n)$ many cycles and edges suffices. The graph $K_{3,n-3}$ shows that at least $(1+c)n$ many cycles and edges are required, for some constant $c>0$. In \\cite{Er71} Erdos suggests that only $n-1$ many cycles and edges are required if we do not require them to be edge-disjoint.\nThe best bound available is due to Buci\\'{c} and Montgomery \\cite{BM22}, who prove that $O(n\\log^*n)$ many cycles and edges suffice, where $\\log^*$ is the iterated logarithm function.\nConlon, Fox, and Sudakov \\cite{CFS14} proved that $O_\\epsilon(n)$ cycles and edges suffice if $G$ has minimum degree at least $\\epsilon n$, for any $\\epsilon>0$.\nSee also [583] for an analogous problem decomposing into paths, and [1017] for decomposing into complete graphs.\nReferences\n\n\n[BM22] Buci\\'C, M. and Montgomery, R., Towards the Erdos-Gallai Cycle Decomposition Conjecture. arXiv:2211.07689 (2022).\n\n[CFS14] Conlon, David and Fox, Jacob and Sudakov, Benny, Cycle packing. Random Structures Algorithms (2014), 608-626.\n\n[Er71] Erdos, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc.\nConf., Oxford, 1969) (1971), 97-109.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1982, "problem_number": "EP-187", "title": "Erdős Problem #187", "statement": "Find the best function $f(d)$ such that, in any 2-colouring of the integers, at least one colour class contains an arithmetic progression with common difference $d$ of length $f(d)$ for infinitely many $d$.", "background": "Originally asked by Cohen. Erdos observed that colouring according to whether $\\{ \\sqrt{2}n\\}<1/2$ or not implies $f(d) \\ll d$ (using the fact that $\\|\\sqrt{2}q\\| \\gg 1/q$ for all $q$, where $\\|x\\|$ is the distance to the nearest integer). Beck \\cite{Be80} has improved this using the probabilistic method, constructing a colouring that shows $f(d)\\leq (1+o(1))\\log_2 d$. Van der Waerden's theorem implies $f(d)\\to \\infty$ is necessary.\nReferences\n\n\n[Be80] Beck, J\\'{o}zsef, A remark concerning arithmetic progressions. J. Combin. Theory Ser. A (1980), 376-379.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1983, "problem_number": "EP-188", "title": "Erdős Problem #188", "statement": "What is the smallest $k$ such that $\\mathbb{R}^2$ can be red/blue coloured with no pair of red points unit distance apart, and no $k$-term arithmetic progression of blue points with distance $1$?", "background": "Erdos, Graham, Montgomery, Rothschild, Spencer, and Straus \\cite{EGMRSS75} proved $k\\geq 5$. Tsaturian \\cite{Ts17} improved this to $k\\geq 6$. Erdos and Graham claim that $k\\leq 10000000$ ('more or less'), but give no proof.\nErdos and Graham asked this with just any $k$-term arithmetic progression in blue (not necessarily with distance $1$), but Alon has pointed out that in fact no such $k$ exists: in any red/blue colouring of the integer points on a line either there are two red points distance $1$ apart, or else the set of blue points and the same set shifted by $1$ cover all integers, and hence by van der Waerden's theorem there are arbitrarily long blue arithmetic progressions.\nIt seems most likely, from context, that Erdos and Graham intended to restrict the blue arithmetic progression to have distance $1$ (although they do not write this restriction in their papers).\nReferences\n\n\n[EGMRSS75] Erdos, P. and Graham, R. L. and Montgomery, P. and\nRothschild, B. L. and Spencer, J. and Straus, E. G., Euclidean {R}amsey theorems. {II}. (1975), 529--557.\n\n[Ts17] Tsaturian, Sergei, A {E}uclidean {R}amsey result in the plane. Electron. J. Combin. (2017), Paper No. 4.35, 9.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1984, "problem_number": "EP-190", "title": "Erdős Problem #190", "statement": "Let $H(k)$ be the smallest $N$ such that in any finite colouring of $\\{1,\\ldots,N\\}$ (into any number of colours) there is always either a monochromatic $k$-term arithmetic progression or a rainbow arithmetic progression (i.e. all elements are different colours). Estimate $H(k)$. Is it true that $ H(k)^{1/k}/k \\to \\infty $ as $k\\to\\infty$?", "background": "This type of problem belongs to 'canonical' Ramsey theory. The existence of $H(k)$ follows from Szemer\\'{e}di's theorem, and it is easy to show that $H(k)^{1/k}\\to\\infty$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1985, "problem_number": "EP-193", "title": "Erdős Problem #193", "statement": "Let $S\\subseteq \\mathbb{Z}^3$ be a finite set and let $A=\\{a_1,a_2,\\ldots,\\}\\subset \\mathbb{Z}^3$ be an infinite $S$-walk, so that $a_{i+1}-a_i\\in S$ for all $i$. Must $A$ contain three collinear points?", "background": "Originally conjectured by Gerver and Ramsey \\cite{GeRa79}, who showed that the answer is yes for $\\mathbb{Z}^2$, and for $\\mathbb{Z}^3$ that the largest number of collinear points can be bounded.\nReferences\n\n\n[GeRa79] Gerver, Joseph L. and Ramsey, L. Thomas, On certain sequences of lattice points. Pacific J. Math. (1979), 357-363.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1986, "problem_number": "EP-195", "title": "Erdős Problem #195", "statement": "What is the largest $k$ such that in any permutation of $\\mathbb{Z}$ there must exist a monotone $k$-term arithmetic progression $x_1<\\cdotsj>k>l$ such that $x_i,x_j,x_k,x_l$ are an arithmetic progression?", "background": "Davis, Entringer, Graham, and Simmons \\cite{DEGS77} have shown that there must exist a monotone 3-term arithmetic progression and need not contain a 5-term arithmetic progression.\nSee also [194] and [195].\nReferences\n\n\n[DEGS77] Davis, J. A. and Entringer, R. C. and Graham, R. L. and\nSimmons, G. J., On permutations containing no long arithmetic progressions. Acta Arith. (1977/78), 81-90.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1988, "problem_number": "EP-197", "title": "Erdős Problem #197", "statement": "Can $\\mathbb{N}$ be partitioned into two sets, each of which can be permuted to avoid monotone 3-term arithmetic progressions?", "background": "If three sets are allowed then this is possible.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1989, "problem_number": "EP-200", "title": "Erdős Problem #200", "statement": "Does the longest arithmetic progression of primes in $\\{1,\\ldots,N\\}$ have length $o(\\log N)$?", "background": "It follows from the prime number theorem that such a progression has length $\\leq(1+o(1))\\log N$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1990, "problem_number": "EP-201", "title": "Erdős Problem #201", "statement": "Let $G_k(N)$ be such that any set of $N$ integers contains a subset of size at least $G_k(N)$ which does not contain a $k$-term arithmetic progression. Determine the size of $G_k(N)$. How does it relate to $R_k(N)$, the size of the largest subset of $\\{1,\\ldots,N\\}$ without a $k$-term arithmetic progression? Is it true that $ \\lim_{N\\to \\infty}\\frac{R_3(N)}{G_3(N)}=1? $ ", "background": "First asked and investigated by Riddell \\cite{Ri69}. It is trivial that $G_k(N)\\leq R_k(N)$, and it is possible that $G_k(N) 0$, $ \\frac{N}{\\exp((\\log N)^{1/2+\\epsilon})} \\ll_\\epsilon f(N) < \\frac{N}{(\\log N)^c} $ for some $c>0$. Erdos believed the lower bound is closer to the truth.\nThese bounds were improved by Croot \\cite{Cr03b} who proved $ \\frac{N}{L(N)^{\\sqrt{2}+o(1)}}< f(N)<\\frac{N}{L(N)^{1/6-o(1)}}, $ where $L(N)=\\exp(\\sqrt{\\log N\\log\\log N})$. These bounds were further improved by Chen \\cite{Ch05} and then by de la Bret\\'{e}che, Ford, and Vandehey \\cite{BFV13} to $ \\frac{N}{L(N)^{1+o(1)}}0$ and large $n$, $ s_{n+1}-s_n \\ll_\\epsilon s_n^{\\epsilon}? $ Is it true that $ s_{n+1}-s_n \\leq (1+o(1))\\frac{\\pi^2}{6}\\frac{\\log s_n}{\\log\\log s_n}? $ ", "background": "Erdos \\cite{Er51} showed that there are infinitely many $n$ such that $ s_{n+1}-s_n > (1+o(1))\\frac{\\pi^2}{6}\\frac{\\log s_n}{\\log\\log s_n}, $ so this bound would be the best possible.\nIn \\cite{Er79} Erdos says perhaps $s_{n+1}-s_n \\ll \\log s_n$, but he is 'very doubtful'.\nFilaseta and Trifonov \\cite{FiTr92} proved an upper bound of $s_n^{1/5+o(1)}$. Pandey \\cite{Pa24} has improved this exponent to $1/5-c$ for some constant $c>0$.\nGranville \\cite{Gr98} showed that $s_{n+1}-s_n\\ll_\\epsilon s_n^\\epsilon$ for all $\\epsilon>0$ follows from the ABC conjecture.\nSee also [489] and [145]. A more general form of this problem is given in [1101].\nReferences\n\n\n[Er51] Erd\"{o}s, P., Some problems and results in elementary number theory. Publ. Math. Debrecen (1951), 103-109.\n\n[Er79] Erdos, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70.\n\n[FiTr92] Filaseta, M. and Trifonov, O., On gaps between squarefree numbers II. J. London Math. Soc. (1992), 215-221.\n\n[Gr98] Granville, Andrew, {$ABC$} allows us to count squarefrees. Internat. Math. Res. Notices (1998), 991--1009.\n\n[Pa24] Pandey, M., Squarefree numbers in short intervals. arXiv:2401.13981 (2024).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1994, "problem_number": "EP-212", "title": "Erdős Problem #212", "statement": "Is there a dense subset of $\\mathbb{R}^2$ such that all pairwise distances are rational?", "background": "Conjectured by Ulam. Erdos believed there cannot be such a set. This problem is discussed in a blogpost by Terence Tao, in which he shows that there cannot be such a set, assuming the Bombieri-Lang conjecture. The same conclusion was independently obtained by Shaffaf \\cite{Sh18}.\nIndeed, Shaffaf and Tao actually proved that such a rational distance set must be contained in a finite union of real algebraic curves. Solymosi and de Zeeuw \\cite{SdZ10} then proved (unconditionally) that a rational distance set contained in a real algebraic curve must be finite, unless the curve contains a line or a circle.\nAscher, Braune, and Turchet \\cite{ABT20} observed that, combined, these facts imply that a rational distance set in general position must be finite (conditional on the Bombieri-Lang conjecture).\nIn \\cite{Er87b} Erdos mentions that Besicovitch conjectured that the limit points of a rational distance set cannot contain arbitrarily large convex sets.\nReferences\n\n\n[ABT20] Ascher, K. and Braune, L. and Turchet, A., The Erdos-Ulam problem, Lang's conjecture, and uniformity. arXiv:1901.02616 (2020).\n\n[Er87b] Erdos, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Si\\'{o}fok, 1985) (1987), 167-177.\n\n[SdZ10] Solymosi, Jozsef and de Zeeuw, Frank, On a question of Erdos and Ulam. Discrete Comput. Geom. (2010), 393-401.\n\n[Sh18] Shaffaf, Jafar, A solution of the Erdos-Ulam problem on rational\ndistance sets assuming the Bombieri-Lang conjecture. Discrete Comput. Geom. (2018), 283-293.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1995, "problem_number": "EP-213", "title": "Erdős Problem #213", "statement": "Let $n\\geq 4$. Are there $n$ points in $\\mathbb{R}^2$, no three on a line and no four on a circle, such that all pairwise distances are integers?", "background": "Anning and Erdos \\cite{AnEr45} proved there cannot exist an infinite such set. Harborth constructed such a set when $n=5$. The best construction to date, due to Kreisel and Kurz \\cite{KK08}, has $n=7$.\nAscher, Braune, and Turchet \\cite{ABT20} have shown that there is a uniform upper bound on the size of such a set, conditional on the Bombieri-Lang conjecture. Greenfeld, Iliopoulou, and Peluse \\cite{GIP24} have shown (unconditionally) that any such set must be very sparse, in that if $S\\subseteq [-N,N]^2$ has no three on a line and no four on a circle, and all pairwise distances integers, then $ \\lvert S\\rvert \\ll (\\log N)^{O(1)}. $ See also [130].\nReferences\n\n\n[ABT20] Ascher, K. and Braune, L. and Turchet, A., The Erdos-Ulam problem, Lang's conjecture, and uniformity. arXiv:1901.02616 (2020).\n\n[AnEr45] Anning, Norman H. and Erdos, Paul, Integral distances. Bull. Amer. Math. Soc. (1945), 598-600.\n\n[GIP24] Greenfeld, R. and Iliopoulou, M. and Peluse, S., On integer distance sets. arXiv:2401.10821 (2024).\n\n[KK08] Kreisel, Tobias and Kurz, Sascha, There are integral heptagons, no three points on a line, on four on a circle. Discrete Comput. Geom. (2008), 786-790.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1996, "problem_number": "EP-217", "title": "Erdős Problem #217", "statement": "For which $n$ are there $n$ points in $\\mathbb{R}^2$, no three on a line and no four on a circle, which determine $n-1$ distinct distances and so that (in some ordering of the distances) the $i$th distance occurs $i$ times?", "background": "An example with $n=4$ is an isosceles triangle with the point in the centre. Erdos originally believed this was impossible for $n\\geq 5$, but Pomerance constructed a set with $n=5$ (see \\cite{Er83c} for a description), and Pal\\'{a}sti has proved such sets exist for all $n\\leq 8$.\nErdos believed this is impossible for all sufficiently large $n$. This would follow from $h(n)\\geq n$ for sufficiently large $n$, where $h(n)$ is as in [98].\nReferences\n\n\n[Er83c] Erdos, Paul, Combinatorial problems in geometry. Math. Chronicle (1983), 35-54.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1997, "problem_number": "EP-218", "title": "Erdős Problem #218", "statement": "Let $d_n=p_{n+1}-p_n$. The set of $n$ such that $d_{n+1}\\geq d_n$ has density $1/2$, and similarly for $d_{n+1}\\leq d_n$. Furthermore, there are infinitely many $n$ such that $d_{n+1}=d_n$.", "background": "In \\cite{Er85c} Erdos also conjectures that $d_n=d_{n+1}=\\cdots=d_{n+k}$ is solvable for every $k$ (which is equivalent to $k$ consecutive primes in arithmetic progression, see [141]).\nReferences\n\n\n[Er85c] Erdos, P., On some of my problems in number theory I would most like to see solved. Number theory (Ootacamund, 1984) (1985), 74-84.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 1998, "problem_number": "EP-222", "title": "Erdős Problem #222", "statement": "Let $n_10$.\nThe sequence of values of $f(n)$ is A109925 on the OEIS.\nSee also [237].\nReferences\n\n\n[Er50] Erd\"{o}s, P., On integers of the form $2^k+p$ and some related problems. Summa Brasil. Math. (1950), 113-123.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[MiWe69] Mientka, Walter E. and Weitzenkamp, Roger C., On {$f$}-plentiful numbers. J. Combinatorial Theory (1969), 374--377.\n\n[Va73] Vaughan, R. C., Some applications of {M}ontgomery's sieve. J. Number Theory (1973), 64--79.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2002, "problem_number": "EP-238", "title": "Erdős Problem #238", "statement": "Let $c_1,c_2>0$. Is it true that, for any sufficiently large $x$, there exist more than $c_1\\log x$ many consecutive primes $\\leq x$ such that the difference between any two is $>c_2$?", "background": "Erdos \\cite{Er49c} proved this is true for any $c_2>0$ if $c_1>0$ is sufficiently small (depending on $c_1$).\nReferences\n\n\n[Er49c] Erdos, P., On some applications of {B}run's method. Acta Univ. Szeged. Sect. Sci. Math. (1949), 57--63.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2003, "problem_number": "EP-241", "title": "Erdős Problem #241", "statement": "Let $f(N)$ be the maximum size of $A\\subseteq \\{1,\\ldots,N\\}$ such that the sums $a+b+c$ with $a,b,c\\in A$ are all distinct (aside from the trivial coincidences). Is it true that $ f(N)\\sim N^{1/3}? $ ", "background": "Originally asked to Erdos by Bose. Bose and Chowla \\cite{BoCh62} provided a construction proving one half of this, namely $ (1+o(1))N^{1/3}\\leq f(N). $ The best upper bound known to date is due to Green \\cite{Gr01}, $ f(N) \\leq ((7/2)^{1/3}+o(1))N^{1/3} $ (note that $(7/2)^{1/3}\\approx 1.519$).\nMore generally, Bose and Chowla conjectured that the maximum size of $A\\subseteq \\{1,\\ldots,N\\}$ with all $r$-fold sums distinct (aside from the trivial coincidences) then $ \\lvert A\\rvert \\sim N^{1/r}. $ This is known only for $r=2$ (see [30]).\nThis is discussed in problem C11 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[BoCh62] Bose, R. C. and Chowla, S., Theorems in the additive theory of numbers. Comment. Math. Helv. (1962/63), 141-147.\n\n[Gr01] Green, Ben, The number of squares and {$B_h[g]$} sets. Acta Arith. (2001), 365-390.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2004, "problem_number": "EP-243", "title": "Erdős Problem #243", "statement": "Let $1\\leq a_10. $ A sequence satisfying the reucrrence $a_n = a_{n-1}^2-a_{n-1}+1$ is known as Sylvester's sequence.\nDuverney \\cite{Du01} proved a weaker version of this problem: if $ \\sum_{n\\geq 0}\\left(\\frac{a_{n+1}}{a_n^2}-1\\right) $ converges then $\\sum \\frac{1}{a_n}$ is rational if and only if $ a_{n}=a_{n-1}^2-a_{n-1}+1 $ for all large $n$.\nReferences\n\n\n[Du01] Duverney, Daniel, Irrationality of fast converging series of rational numbers. J. Math. Sci. Univ. Tokyo (2001), 275--316.\n\n[ErSt64] Erdos, P. and Straus, E. G., On the irrationality of certain {A}hmes series. J. Indian Math. Soc. (N.S.) (1964), 129--133.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2005, "problem_number": "EP-244", "title": "Erdős Problem #244", "statement": "Let $C>1$. Does the set of integers of the form $p+\\lfloor C^k\\rfloor$, for some prime $p$ and $k\\geq 0$, have density $>0$?", "background": "Originally asked to Erdos by Kalm\\'{a}r. Erdos believed the answer is yes. Romanoff \\cite{Ro34} proved that the answer is yes if $C$ is an integer.\nDing \\cite{Di25} has proved that this is true for almost all $C>1$.\nReferences\n\n\n[Di25] Y. Ding, On a Romanoff type problem of Erdos and Kalm\\'{a}r. arXiv:2503.22700 (2025).\n\n[Ro34] Romanoff, N. P., \"{U}ber einige S\"Atze der additiven Zahlentheorie. Math. Ann. (1934), 668-678.\n\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2006, "problem_number": "EP-247", "title": "Erdős Problem #247", "statement": "Let $1\\leq a_1cn^2$ then $\\sum_{n=1}^\\infty \\frac{1}{2^{a_n}}$ is not the root of any quadratic polynomial'.\nThis problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.\nReferences\n\n\n[Er75c] Erdos, P., Some problems and results on the irrationality of the sum of infinite series. J. Math. Sci. (1975), 1-7 (1976).\n\n[Er88c] Erd\"{o}s, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109.\n\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2007, "problem_number": "EP-249", "title": "Erdős Problem #249", "statement": "Is $ \\sum_n \\frac{\\phi(n)}{2^n} $ irrational? Here $\\phi$ is the Euler totient function.", "background": "The decimal expansion of this sum is A256936 on the OEIS.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2008, "problem_number": "EP-251", "title": "Erdős Problem #251", "statement": "Is $ \\sum \\frac{p_n}{2^n} $ irrational? (Here $p_n$ is the $n$th prime.)", "background": "Erdos \\cite{Er58b} proved that $\\sum \\frac{p_n^k}{n!}$ is irrational for every $k\\geq 1$.\nIn \\cite{Er88c} he further conjectures that $\\sum \\frac{p_n^k}{2^n}$ is irrational for every $k$, and that if $g_n\\geq 2$ and $g_n=o(p_n)$ then $ \\sum_{n=1}^\\infty \\frac{p_n}{g_1\\cdots g_n} $ is irrational. (The example $g_n=p_n+1$ shows that some condition on the growth of the $g_n$ is necessary here.)\nThe decimal expansion of this sum is A098990 on the OEIS.\nReferences\n\n\n[Er58b] Erdos, Paul, Sur certaines s\\'{e}ries \\`a{} valeur irrationnelle. Enseign. Math. (2) (1958), 93--100.\n\n[Er88c] Erd\"{o}s, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2009, "problem_number": "EP-252", "title": "Erdős Problem #252", "statement": "Let $k\\geq 1$ and $\\sigma_k(n)=\\sum_{d\\mid n}d^k$. Is $ \\sum \\frac{\\sigma_k(n)}{n!} $ irrational?", "background": "This is known now for $1\\leq k\\leq 4$. The cases $k=1,2$ are reasonably straightforward, as observed by Erdos \\cite{Er52}. The case $k=3$ was proved independently by Schlage-Puchta \\cite{ScPu06} and Friedlander, Luca, and Stoiciu \\cite{FLC07}. The case $k=4$ was proved by Pratt \\cite{Pr22}.\nIt is known that this sum is irrational for all $k\\geq 1$ conditional on either Schinzel's conjecture (Schlage-Puchta \\cite{ScPu06}) or the prime tuples conjecture (Friedlander, Luca, and Stoiciu \\cite{FLC07}).\nThis is discussed in problem B14 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Er52] Erdos, P., Problem 4493. Amer. Math. Monthly (1952), 557-558.\n\n[FLC07] Friedlander, J. B. and Luca, F. and Stoiciu, M., On the irrationality of a divisor function series. Integers (2007).\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Pr22] Pratt, K., The irrationality of a divisor function series of Erdos and Kac. arXiv:2209.11124 (2022).\n\n[ScPu06] Schlage-Puchta, J. C., The irrationality of a number theoretical series. Ramanujan J. (2006), 455-460.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2010, "problem_number": "EP-254", "title": "Erdős Problem #254", "statement": "Let $A\\subseteq \\mathbb{N}$ be such that $ \\lvert A\\cap [1,2x]\\rvert -\\lvert A\\cap [1,x]\\rvert \\to \\infty\\textrm{ as }x\\to \\infty $ and $ \\sum_{n\\in A} \\{ \\theta n\\}=\\infty $ for every $\\theta\\in (0,1)$, where $\\{x\\}$ is the distance of $x$ from the nearest integer. Then every sufficiently large integer is the sum of distinct elements of $A$.", "background": "Cassels \\cite{Ca60} proved this under the alternative hypotheses $ \\lim \\frac{\\lvert A\\cap [1,2x]\\rvert -\\lvert A\\cap [1,x]\\rvert}{\\log\\log x}=\\infty $ and $ \\sum_{n\\in A} \\{ \\theta n\\}^2=\\infty $ for every $\\theta\\in (0,1)$.\nReferences\n\n\n[Ca60] Cassels, J. W. S., On the representation of integers as the sums of distinct summands taken from a fixed set. Acta Sci. Math. (Szeged) (1960), 111-124.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2011, "problem_number": "EP-256", "title": "Erdős Problem #256", "statement": "Let $n\\geq 1$ and $f(n)$ be maximal such that for any integers $1\\leq a_1\\leq \\cdots \\leq a_n$ we have $ \\max_{\\lvert z\\rvert=1}\\left\\lvert \\prod_{i}(1-z^{a_i})\\right\\rvert\\geq f(n). $ Estimate $f(n)$ - in particular, is it true that there exists some constant $c>0$ such that $ \\log f(n) \\gg n^c? $ ", "background": "Erdos and Szekeres \\cite{ErSz59} proved that $\\lim f(n)^{1/n}=1$ and $f(n)>\\sqrt{2n}$. Erdos proved an upper bound of $\\log f(n) \\ll n^{1-c}$ for some constant $c>0$ with probabilistic methods. Atkinson \\cite{At61} showed that $\\log f(n) \\ll n^{1/2}\\log n$.\nThis was improved to $ \\log f(n) \\ll n^{1/3}(\\log n)^{4/3} $ by Odlyzko \\cite{Od82}.\nIf we denote by $f^*(n)$ the analogous quantity with the assumption that $a_1<\\cdots0 $ for some $\\epsilon>0$ then the above folklore result implies that $a_n$ is such an irrationality sequence.\nReferences\n\n\n[KoTa24] Kova\\vC, V. and Tao T., On several irrationality problems for Ahmes series. arXiv:2406.17593 (2024).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2017, "problem_number": "EP-264", "title": "Erdős Problem #264", "statement": "Let $a_n$ be a sequence of positive integers such that for every bounded sequence of integers $b_n$ (with $a_n+b_n\neq 0$ and $b_n\neq 0$ for all $n$) the sum $ \\sum \\frac{1}{a_n+b_n} $ is irrational. Are $a_n=2^n$ or $a_n=n!$ examples of such a sequence?", "background": "A possible definition of an 'irrationality sequence' (see also [262] and [263]). One example is $a_n=2^{2^n}$. In \\cite{ErGr80} they also ask whether such a sequence can have polynomial growth, but Erdos later retracted this in \\cite{Er88c}, claiming 'It is not hard to show that it cannot increase slower than exponentially'.\nKova\\v{c} and Tao \\cite{KoTa24} have proved that $2^n$ is not such an irrationality sequence. More generally, they prove that any strictly increasing sequence of positive integers such that $\\sum\\frac{1}{a_n}$ converges and $ \\liminf \\left(a_n^2\\sum_{k>n}\\frac{1}{a_k^2}\\right) >0 $ is not such an irrationality sequence. In particular, any strictly increasing sequence with $\\limsup a_{n+1}/a_n <\\infty$ is not such an irrationality sequence.\nOn the other hand, Kova\\v{c} and Tao do prove that for any function $F$ with $\\lim F(n+1)/F(n)=\\infty$ there exists such an irrationality sequence with $a_n\\sim F(n)$.\nReferences\n\n\n[Er88c] Erd\"{o}s, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109.\n\n[ErGr80] Erdos, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).\n\n[KoTa24] Kova\\vC, V. and Tao T., On several irrationality problems for Ahmes series. arXiv:2406.17593 (2024).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2018, "problem_number": "EP-265", "title": "Erdős Problem #265", "statement": "Let $1\\leq a_11$.\nIt remains open whether one can achieve $ \\limsup a_n^{1/2^n}>1. $ A folklore result states that $\\sum \\frac{1}{a_n}$ is irrational whenever $\\lim a_n^{1/2^n}=\\infty$, and hence such a sequence cannot grow faster than doubly exponentially - the remaining question is the precise exponent possible.\nReferences\n\n\n[KoTa24] Kova\\vC, V. and Tao T., On several irrationality problems for Ahmes series. arXiv:2406.17593 (2024).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2019, "problem_number": "EP-267", "title": "Erdős Problem #267", "statement": "Let $F_1=F_2=1$ and $F_{n+1}=F_n+F_{n-1}$ be the Fibonacci sequence. Let $n_11$. Must $ \\sum_k\\frac{1}{F_{n_k}} $ be irrational?", "background": "It may be sufficient to have $n_k/k\\to \\infty$. Good \\cite{Go74} and Bicknell and Hoggatt \\cite{BiHo76} have shown that $\\sum \\frac{1}{F_{2^n}}$ is irrational - in fact, $ \\sum \\frac{1}{F_{2^n}}=\\frac{7-\\sqrt{5}}{2}. $ Badea \\cite{Ba87} proved that $\\sum \\frac{1}{F_{2^n+1}}$ is irrational.\nThe sum $\\sum \\frac{1}{F_n}$ itself was proved to be irrational by Andr\\'{e}-Jeannin \\cite{An89}.\nThe main problem has been proved for $c\\geq 2$ by Badea \\cite{Ba93}. It remains open for $10$, $a_k\\leq (\\frac{1}{2}+\\epsilon)k^2$ for all sufficiently large $k$. van Doorn and Sothanaphan have noted in the comment section that Moy's proof can be upgraded to give a fully explicit result of $ a_k\\leq \\frac{(k-1)(k+2)}{2}+n $ for all $k\\geq 0$.\nIn general, sequences which begin with some initial segment and thereafter are continued in a greedy fashion to avoid three-term arithmetic progressions are known as Stanley sequences.\nReferences\n\n\n[Li90] S. Lindhurst, An investigation of several interesting sets of numbers generated by the greedy\nalgorithm. Senior thesis at Princeton University (1990).\n\n[Mo11] Moy, Richard A., On the growth of the counting function of Stanley sequences. Discrete Math. (2011), 560-562.\n\n[OdSt78] A. Odlyzko and R. Stanley, Some curious sequences constructed with the greedy algorithm. Bell Laboratories internal memorandum (1978).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2022, "problem_number": "EP-272", "title": "Erdős Problem #272", "statement": "Let $N\\geq 1$. What is the largest $t$ such that there are $A_1,\\ldots,A_t\\subseteq \\{1,\\ldots,N\\}$ with $A_i\\cap A_j$ a non-empty arithmetic progression for all $i\neq j$?", "background": "Simonovits and S\\'{o}s \\cite{SiSo81} have shown that $t\\ll N^2$.\nErdos and Graham asked whether the maximal $t$ is achieved when we take the $A_i$ to be all arithmetic progressions in $\\{1,\\ldots,N\\}$ containing some fixed element, 'presumably the integer $\\lfloor N/2\\rfloor$'. This was disproved by Simonovits and S\\'{o}s \\cite{SiSo81}, who observed that taking all sets containing at most $3$ elements, containing some fixed element, produces $\\binom{N}{2}+1$ many such sets, which is asymptotically greater than the number of arithmetic progressions containing a fixed element, which is $\\sim \\frac{\\pi^2}{24}N^2$.\nIf we drop the non-empty requirement then Graham, Simonovits, and S\\'{o}s \\cite{GSS80} have shown that $ t\\leq \\binom{N}{3}+\\binom{N}{2}+\\binom{N}{1}+1 $ and this is best possible.\nSzabo \\cite{Sz99} proved that the maximal such $t$ is equal to $ \\frac{N^2}{2}+O(N^{5/3}(\\log N)^3), $ resolving the asymptotic question. On the other hand, Szabo showed that the conjecture of Simonovits and S\\'{o}s that $\\binom{n}{2}+1$ is best possible is false, giving a construction which yields $ t \\geq \\binom{N}{2}+\\left\\lfloor\\frac{N-1}{4}\\right\\rfloor+1. $ Szabo conjectures that the asymptotic $t=\\binom{N}{2}+O(N)$ holds, and that in any extremal example there is an integer contained in all sets.\nReferences\n\n\n[GSS80] Graham, R. L. and Simonovits, M. and S\\'{o}s, V. T., A note on the intersection properties of subsets of integers. J. Combin. Theory Ser. A (1980), 106-110.\n\n[SiSo81] Simonovits, Mikl\\'{o}s and S\\'{o}s, Vera T., Intersection properties of subsets of integers. European J. Combin. (1981), 363-372.\n\n[Sz99] Szab\\'o, Tibor, Intersection properties of subsets of integers. European J. Combin. (1999), 429--444.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2023, "problem_number": "EP-273", "title": "Erdős Problem #273", "statement": "Is there a covering system all of whose moduli are of the form $p-1$ for some primes $p\\geq 5$?", "background": "Selfridge has found an example using divisors of $360$ if $p=3$ is allowed.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2024, "problem_number": "EP-274", "title": "Erdős Problem #274", "statement": "If $G$ is a group then can there exist an exact covering of $G$ by more than one cosets of different sizes? (i.e. each element is contained in exactly one of the cosets)", "background": "A question of Herzog and Sch\"{onheim}, who conjectured more generally that if $G$ is any (not necessarily finite) group and $a_1G_1,\\ldots,a_kG_k$ are finitely many cosets of subgroups of $G$ with distinct indices $[G:G_i]$ then the $a_iG_i$ cannot form a partition of $G$.\nThis conjecture was proved in the case when all the $G_i$ are subnormal in $G$ by Sun \\cite{Su04}. In particular if $G$ is abelian (which was the special case asked about in \\cite{Er77c} and \\cite{ErGr80}) the answer to the original question is no.\nMargolis and Schnabel \\cite{MaSc19} proved this conjecture for all groups $G$ of size $<1440$.\nReferences\n\n\n[Er77c] Erdos, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ.,\nNew York, 1976) (1977), 43-72.\n\n[ErGr80] Erdos, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).\n\n[MaSc19] Margolis, Leo and Schnabel, Ofir, The {H}erzog-{S}ch\"onheim conjecture for small groups and\nharmonic subgroups. Beitr. Algebra Geom. (2019), 399--418.\n\n[Su04] Sun, Zhi-Wei, On the {H}erzog-{S}ch\"onheim conjecture for uniform covers of\ngroups. J. Algebra (2004), 153--175.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2025, "problem_number": "EP-276", "title": "Erdős Problem #276", "statement": "Is there an infinite Lucas sequence $a_0,a_1,\\ldots$ where $a_{n+2}=a_{n+1}+a_n$ for $n\\geq 0$ such that all $a_k$ are composite, and yet no integer has a common factor with every term of the sequence?", "background": "Whether such a composite Lucas sequence even exists was open for a while, but using covering systems Graham \\cite{Gr64} showed that $ a_0 = 1786772701928802632268715130455793 $ and $ a_1 = 1059683225053915111058165141686995 $ generate such a sequence. This problem asks whether one can have a composite Lucas sequence without 'an underlying system of covering congruences responsible'.\nThis problem has been 'conjecturally solved' by Ismailescu and Son \\cite{IsSo14}, in that they provide an explicit infinite Lucas sequence in which all the terms are composite, and believe that no covering system is responsible for this. See the comment by van Doorn below for more details.\nSee also [1113] for another problem in which the question is whether covering systems are always responsible.\nReferences\n\n\n[Gr64] Graham, R. L., A Fibonacci-Like Sequence of Composite Numbers. Math. Mag. (1964), 322-324.\n\n[IsSo14] Ismailescu, Dan and Son, Jaesung, A new kind of {F}ibonacci-like sequence of composite numbers. J. Integer Seq. (2014), Article 14.8.2, 9.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2026, "problem_number": "EP-278", "title": "Erdős Problem #278", "statement": "Let $A=\\{n_1<\\cdots0$ there exists some $k$ such that, for every choice of congruence classes $a_i$, the density of integers not satisfying any of the congruences $a_i\\pmod{n_i}$ for $1\\leq i\\leq k$ is less than $\\epsilon$?", "background": "The latter condition is clearly sufficient, the problem is if it's also necessary. The assumption implies $\\sum \\frac{1}{n_i}=\\infty$. If the $n_i$ are pairwise relatively prime then it is sufficient that $\\sum \\frac{1}{n_i}=\\infty$.\nThis is true - a proof is given in the comments by Somani (using ChatGPT).\nAn alternative elementary proof was noted by KoishiChan in the comments: let $\\delta$ be the lower density of the set of integers divisible by some $n\\in A$, and $\\delta_k$ be the density of the set of integers divisible by at least one of $n_1,\\ldots,n_k$. A theorem of Davenport and Erdos \\cite{DaEr36} states that $ \\delta = \\lim_{k\\to \\infty}\\delta_k. $ In the present case $\\delta=1$ and hence for every $\\epsilon>0$ there exists $k$ such that $\\delta_k>1-\\epsilon$. In other words, the density of those integers not satisfying $a_i\\equiv 0\\pmod{n_i}$ for $1\\leq i\\leq k$ is $<\\epsilon$. By a theorem of Rogers (which first appeared in print in Chapter V.3 of the book of Halberstam and Roth \\cite{HaRo66}) the density of those integers not satisfying any of the congruences $a_i\\pmod{n_i}$ for $1\\leq i\\leq k$ is maximised when $a_i\\equiv 0$, which concludes the proof.\nGiven that both Rogers' result and the Davenport-Erdos theorem mentioned above must have been very familiar to Erdos in 1980, it is strange that this natural argument was overlooked.\nReferences\n\n\n[DaEr36] Davenport, H. and Erdos, P., On sequences of positive integers. Acta Arithmetica (1936), 147-151.\n\n[HaRo66] Halberstam, H. and Roth, K. F., Sequences. {V}ol. {I}. (1966), xx+291.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2029, "problem_number": "EP-282", "title": "Erdős Problem #282", "statement": "Let $A\\subseteq \\mathbb{N}$ be an infinite set and consider the following greedy algorithm for a rational $x\\in (0,1)$: choose the minimal $n\\in A$ such that $n\\geq 1/x$ and repeat with $x$ replaced by $x-\\frac{1}{n}$. If this terminates after finitely many steps then this produces a representation of $x$ as the sum of distinct unit fractions with denominators from $A$.\nDoes this process always terminate if $x$ has odd denominator and $A$ is the set of odd numbers? More generally, for which pairs $x$ and $A$ does this process terminate?", "background": "In 1202 Fibonacci observed that this process terminates for any $x$ when $A=\\mathbb{N}$. The problem when $A$ is the set of odd numbers is due to Stein.\nGraham \\cite{Gr64b} has shown that $\\frac{m}{n}$ is the sum of distinct unit fractions with denominators $\\equiv a\\pmod{d}$ if and only if $ \\left(\\frac{n}{(n,(a,d))},\\frac{d}{(a,d)}\\right)=1. $ Does the greedy algorithm always terminate in such cases?\nGraham \\cite{Gr64c} has also shown that $x$ is the sum of distinct unit fractions with square denominators if and only if $x\\in [0,\\pi^2/6-1)\\cup [1,\\pi^2/6)$. Does the greedy algorithm for this always terminate? Erdos and Graham believe not - indeed, perhaps it fails to terminate almost always.\nSee also [206].\nReferences\n\n\n[Gr64b] Graham, R. L., On finite sums of unit fractions. Proc. London Math. Soc. (3) (1964), 193-207.\n\n[Gr64c] Graham, R. L., On finite sums of reciprocals of distinct nth powers. Pacific J. Math. (1964), 85-92.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2030, "problem_number": "EP-283", "title": "Erdős Problem #283", "statement": "Let $p:\\mathbb{Z}\\to \\mathbb{Z}$ be a polynomial whose leading coefficient is positive and such that there exists no $d\\geq 2$ with $d\\mid p(n)$ for all $n\\geq 1$. Is it true that, for all sufficiently large $m$, there exist integers $1\\leq n_1<\\cdots 0$ (provided $m$ is taken sufficiently large depending on $\\alpha$).\nCassels \\cite{Ca60} has proved that these conditions on the polynomial imply every sufficiently large integer is the sum of $p(n_i)$ with distinct $n_i$. Burr has proved this if $p(x)=x^k$ with $k\\geq 1$ and if we allow $n_i=n_j$.\nAlekseyev \\cite{Al19} has proved this when $p(x)=x^2$, for all $m>8542$. For example, $ 1=\\frac{1}{2}+\\frac{1}{4}+\\frac{1}{6}+\\frac{1}{12} $ and $ 200 = 2^2+4^2+6^2+12^2. $ van Doorn \\cite{vD25} has investigated the question of what 'sufficiently large' means for $p(x)=x$. van Doorn has also proved the original conjecture for many linear and quadratic polynomials, for example $p(x)=x+5$ or $p(x)=x^2+100$ - see the comments section.\nReferences\n\n\n[Al19] Alekseyev, Max A., On partitions into squares of distinct integers whose\nreciprocals sum to 1. (2019), 213--221.\n\n[Ca60] Cassels, J. W. S., On the representation of integers as the sums of distinct summands taken from a fixed set. Acta Sci. Math. (Szeged) (1960), 111-124.\n\n[Gr63] Graham, R. L., A theorem on partitions. J. Austral. Math. Soc. (1963), 435-441.\n\n[vD25] W. van Doorn, Partitions with prescribed sum of rationals: asymptotic bounds. arXiv:2502.02200 (2025).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2031, "problem_number": "EP-288", "title": "Erdős Problem #288", "statement": "Is it true that there are only finitely many pairs of intervals $I_1,I_2$ such that $ \\sum_{n_1\\in I_1}\\frac{1}{n_1}+\\sum_{n_2\\in I_2}\\frac{1}{n_2}\\in \\mathbb{N}? $ ", "background": "For example, $ \\frac{1}{3}+\\frac{1}{4}+\\frac{1}{5}+\\frac{1}{6}+\\frac{1}{20}=1. $ This is still open even if $\\lvert I_2\\rvert=1$. It is perhaps true with two intervals replaced by any $k$ intervals.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2032, "problem_number": "EP-289", "title": "Erdős Problem #289", "statement": "Is it true that, for all sufficiently large $k$, there exist finite intervals $I_1,\\ldots,I_k\\subset \\mathbb{N}$, distinct, not overlapping or adjacent, with $\\lvert I_i\\rvert \\geq 2$ for $1\\leq i\\leq k$ such that $ 1=\\sum_{i=1}^k \\sum_{n\\in I_i}\\frac{1}{n}? $ ", "background": "Erdos and Graham posed this in \\cite{ErGr80} without the stipulation the intervals be distinct, non-overlapping, or adjacent, but Kovac in the comments has provided a simple argument showing that it is easily possible without this restriction, and likely \\cite{ErGr80} just forgot to mention this natural restriction.\nAs an example representing $2$ rather than $1$, Hickerson and Montgomery, in the solution to AMS Monthly problem E2689 proposed by Hahn, found $ 2=\\sum_{i=1}^5 \\sum_{n\\in I_i}\\frac{1}{n} $ where $I_1=[2,7]$, $I_2=[9,10]$, $I_3=[17,18]$, $I_4=[34,35]$, and $I_5=[84,85]$.\nReferences\n\n\n[ErGr80] Erdos, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2033, "problem_number": "EP-291", "title": "Erdős Problem #291", "statement": "Let $n\\geq 1$ and define $L_n$ to be the least common multiple of $\\{1,\\ldots,n\\}$ and $a_n$ by $ \\sum_{1\\leq k\\leq n}\\frac{1}{k}=\\frac{a_n}{L_n}. $ Is it true that $(a_n,L_n)=1$ and $(a_n,L_n)>1$ both occur for infinitely many $n$?", "background": "Steinerberger has observed that the answer to the second question is trivially yes: for example, any $n$ which begins with a $2$ in base $3$ has $3\\mid (a_n,L_n)$.\nMore generally, if the leading digit of $n$ in base $p$ is $p-1$ then $p\\mid (a_n,L_n)$. There is in fact a necessary and sufficient condition: a prime $p\\leq n$ divides $(a_n,L_n)$ if and only if $p$ divides the numerator of $1+\\cdots+\\frac{1}{k}$, where $k$ is the leading digit of $n$ in base $p$. This can be seen by writing $ a_n = \\frac{L_n}{1}+\\cdots+\\frac{L_n}{n} $ and observing that the right-hand side is congruent to $1+\\cdots+1/k$ modulo $p$. (The previous claim about $p-1$ follows immediately from Wolstenholme's theorem.)\nThis leads to a heuristic prediction (see for example a preprint of Shiu \\cite{Sh16}) of $\\asymp\\frac{x}{\\log x}$ for the number of $n\\in [1,x]$ such that $(a_n,L_n)=1$. In particular, there should be infinitely many $n$, but the set of such $n$ should have density zero. Unfortunately this heuristic is difficult to turn into a proof.\nWu and Yan \\cite{WuYa22} have proved, conditional on $\\frac{1}{\\log p}$ being linearly independent over $\\mathbb{Q}$ for any finite collection of primes $p$ (itself a consequence of Schanuel's conjecture), that the set of $n$ for which $(a_n,L_n)>1$ has upper density $1$.\nReferences\n\n\n[Sh16] P. Shiu, The denominators of harmonic numbers. arXiv:1607.02863 (2016).\n\n[WuYa22] Wu, Bing-Ling and Yan, Xiao-Hui, On the denominators of harmonic numbers. {IV}. C. R. Math. Acad. Sci. Paris (2022), 53--57.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2034, "problem_number": "EP-293", "title": "Erdős Problem #293", "statement": "Let $k\\geq 1$ and let $v(k)$ be the minimal integer which does not appear as some $n_i$ in a solution to $ 1=\\frac{1}{n_1}+\\cdots+\\frac{1}{n_k} $ with $1\\leq n_1<\\cdots 0$, and noted a close connection to [304]. In particular, if $N(b)\\ll \\log\\log b$ as in [304] then it is likely the methods of \\cite{vDTa25b} prove $v(k) \\geq e^{e^{ck}}$ for some $c>0$.\nReferences\n\n\n[BlEr75] Bleicher, M. N. and Erdos, P., The number of distinct subsums of $\\sum \\sb{1}\\spN\\,1/i$. Math. Comp. (1975), 29-42.\n\n[vDTa25b] W. van Doorn and Q. Tang, The smallest denominator not contained in a unit fraction decomposition of 1 with fixed length. arXiv:2512.22083 (2025).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2035, "problem_number": "EP-295", "title": "Erdős Problem #295", "statement": "Let $N\\geq 1$ and let $k(N)$ denote the smallest $k$ such that there exist $N\\leq n_1<\\cdots 0$ such that $ -c < k(N)-(e-1)N \\ll \\frac{N}{\\log N}. $ \nReferences\n\n\n[ErSt71b] Erdos, P. and Straus, E. G., Solution to Problem. Amer. Math. Monthly (1971), 302-303.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2036, "problem_number": "EP-301", "title": "Erdős Problem #301", "statement": "Let $f(N)$ be the size of the largest $A\\subseteq \\{1,\\ldots,N\\}$ such that there are no solutions to $ \\frac{1}{a}= \\frac{1}{b_1}+\\cdots+\\frac{1}{b_k} $ with distinct $a,b_1,\\ldots,b_k\\in A$?\nEstimate $f(N)$. In particular, is it true that $f(N)=(\\tfrac{1}{2}+o(1))N$?", "background": "The example $A=(N/2,N]\\cap \\mathbb{N}$ shows that $f(N)\\geq N/2$.\nWouter van Doorn has given an elementary argument that proves $ f(N)\\leq (25/28+o(1))N. $ Indeed, consider the sets $S_a=\\{2a,3a,4a,6a,12a\\}\\cap [1,N]$ as $a$ ranges over all integers of the form $8^b9^cd$ with $(d,6)=1$. All such $S_a$ are disjoint and, if $A$ has no solutions to the given equation, then $A$ must omit at least two elements of $S_a$ when $a\\leq N/12$ and at least one element of $S_a$ when $N/120}$ with $b$ squarefree. Are there integers $10$.\nReferences\n\n\n[ErGr80] Erdos, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2041, "problem_number": "EP-312", "title": "Erdős Problem #312", "statement": "Does there exist some $c>0$ such that, for any $K>1$, whenever $A$ is a sufficiently large finite multiset of positive integers with $\\sum_{n\\in A}\\frac{1}{n}>K$ there exists some $S\\subseteq A$ such that $ 1-e^{-cK} < \\sum_{n\\in S}\\frac{1}{n}\\leq 1? $ ", "background": "Erdos and Graham knew this with $e^{-cK}$ replaced by $c/K^2$.\n\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2042, "problem_number": "EP-313", "title": "Erdős Problem #313", "statement": "Are there infinitely many solutions to $ \\frac{1}{p_1}+\\cdots+\\frac{1}{p_k}=1-\\frac{1}{m}, $ where $m\\geq 2$ is an integer and $p_1<\\cdots0$ such that for every $n\\geq 1$ there exists some $\\delta_k\\in \\{-1,0,1\\}$ for $1\\leq k\\leq n$ with $ 0< \\left\\lvert \\sum_{1\\leq k\\leq n}\\frac{\\delta_k}{k}\\right\\rvert < \\frac{c}{2^n}? $ Is it true that for sufficiently large $n$, for any $\\delta_k\\in \\{-1,0,1\\}$, $ \\left\\lvert \\sum_{1\\leq k\\leq n}\\frac{\\delta_k}{k}\\right\\rvert > \\frac{1}{[1,\\ldots,n]} $ whenever the left-hand side is not zero?", "background": "Inequality is obvious for the second claim, the problem is strict inequality. This fails for small $n$, for example $ \\frac{1}{2}-\\frac{1}{3}-\\frac{1}{4}=-\\frac{1}{12}. $ Arguments of Kovac and van Doorn in the comment section prove a weak version of the first question, with an upper bound of $ 2^{-n\\frac{(\\log\\log\\log n)^{1+o(1)}}{\\log n}}, $ and van Doorn gives a heuristic that suggests this may be the true order of magnitude.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2044, "problem_number": "EP-318", "title": "Erdős Problem #318", "statement": "Let $A\\subseteq \\mathbb{N}$ be an infinite arithmetic progression and $f:A\\to \\{-1,1\\}$ be a non-constant function. Must there exist a finite non-empty $S\\subset A$ such that $ \\sum_{n\\in S}\\frac{f(n)}{n}=0? $ What about if $A$ is an arbitrary set of positive density? What if $A$ is the set of squares excluding $1$?", "background": "Erdos and Straus \\cite{ErSt75} proved this when $A=\\mathbb{N}$. Sattler \\cite{Sa75} proved this when $A$ is the set of odd numbers. For the squares $1$ must be excluded or the result is trivially false, since $ \\sum_{k\\geq 2}\\frac{1}{k^2}<1. $ This is false for some sets $A$ of positive density - indeed, it fails for any set $A$ containing exactly one even number. (Sattler \\cite{Sa82} credits this observation to Erdos, who presumably found this after \\cite{ErGr80}.)\nSattler \\cite{Sa82b} proved the answer to the original question is yes, in that any arithmetic progression has this property.\nThe final question of the set of squares excluding $1$ appears to be open - Sattler announced a proof in \\cite{Sa82} and \\cite{Sa82b}, but this never appeared.\nReferences\n\n\n[ErGr80] Erdos, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).\n\n[ErSt75] Erdos, P. and Straus, E. G., Solution to Problem 387. Nieuw Arch. Wisk. (1975), 183.\n\n[Sa75] Sattler, R., Solution to Problem 387. Nieuw Arch. Wisk. (1975), 184-189.\n\n[Sa82] Sattler, R., On {E}rd\\H{o}s property {${\\rm P}\\sb{1}$}\\ for the sequence of\nsquarefree numbers. Nederl. Akad. Wetensch. Indag. Math. (1982), 341--346.\n\n[Sa82b] Sattler, R., On {E}rd\\H{o}s property {${\\rm P}\\sb{1}$}\\ for the arithmetical\nsequence. Nederl. Akad. Wetensch. Indag. Math. (1982), 347--352.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2045, "problem_number": "EP-319", "title": "Erdős Problem #319", "statement": "What is the size of the largest $A\\subseteq \\{1,\\ldots,N\\}$ such that there is a function $\\delta:A\\to \\{-1,1\\}$ such that $ \\sum_{n\\in A}\\frac{\\delta_n}{n}=0 $ and $ \\sum_{n\\in A'}\\frac{\\delta_n}{n}\neq 0 $ for all non-empty $A'\\subsetneq A$?", "background": "Adenwalla has observed that a lower bound of $ \\lvert A\\rvert\\geq (1-\\tfrac{1}{e}+o(1))N $ follows from the main result of Croot \\cite{Cr01}, which states that there exists some set of integers $B\\subset [(\\frac{1}{e}-o(1))N,N]$ such that $\\sum_{b\\in B}\\frac{1}{b}=1$. Since the sum of $\\frac{1}{m}$ for $m\\in [c_1N,c_2N]$ is asymptotic to $\\log(c_2/c_1)$ we must have $\\lvert B\\rvert \\geq (1-\\tfrac{1}{e}+o(1))N$.\nWe may then let $A=B\\cup\\{1\\}$ and choose $\\delta(n)=-1$ for all $n\\in B$ and $\\delta(1)=1$.\nThis problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.\nReferences\n\n\n[Cr01] Croot, III, Ernest S., On unit fractions with denominators in short intervals. Acta Arith. (2001), 99-114.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2046, "problem_number": "EP-320", "title": "Erdős Problem #320", "statement": "Let $S(N)$ count the number of distinct sums of the form $\\sum_{n\\in A}\\frac{1}{n}$ for $A\\subseteq \\{1,\\ldots,N\\}$. Estimate $S(N)$.", "background": "Bleicher and Erdos \\cite{BlEr75} proved the lower bound $ \\log S(N)\\geq \\frac{N}{\\log N}\\left(\\log 2\\prod_{i=3}^k\\log_iN\\right), $ valid for $k\\geq 4$ and $\\log_kN\\geq k$, and also \\cite{BlEr76b} proved the upper bound $ \\log S(N)\\leq \\frac{N}{\\log N}\\left(\\log_r N \\prod_{i=3}^r \\log_iN\\right), $ valid for $r\\geq 1$ and $\\log_{2r}N\\geq 1$. (In these bounds $\\log_in$ denotes the $i$-fold iterated logarithm.)\nBettin, Greni\\'{e}, Molteni, and Sanna \\cite{BGMS25} improved the lower bound to $ \\log S(N) \\geq \\frac{N}{\\log N}\\left(2\\log 2\\left(1-\\frac{3/2}{\\log_kN}\\right)\\prod_{i=3}^k\\log_iN\\right), $ valid for $k\\geq 4$ and $\\log_kN\\geq 3/2$. (In particular this goes to infinity faster than the lower bound of Bleicher and Erdos.)\nSee also [321].\nReferences\n\n\n[BGMS25] S. Bettin, L. Greni\\'{e}, G. Molteni, and C. Sanna, A lower bound for the number of Egyptian fractions. arXiv:2509.10030 (2025).\n\n[BlEr75] Bleicher, M. N. and Erdos, P., The number of distinct subsums of $\\sum \\sb{1}\\spN\\,1/i$. Math. Comp. (1975), 29-42.\n\n[BlEr76b] Bleicher, Michael N. and Erdos, Paul, Denominators of Egyptian fractions. II. Illinois J. Math. (1976), 598-613.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2047, "problem_number": "EP-321", "title": "Erdős Problem #321", "statement": "What is the size of the largest $A\\subseteq \\{1,\\ldots,N\\}$ such that all sums $\\sum_{n\\in S}\\frac{1}{n}$ are distinct for $S\\subseteq A$?", "background": "Let $R(N)$ be the maximal such size. Results of Bleicher and Erdos from \\cite{BlEr75} and \\cite{BlEr76b} imply that $ \\frac{N}{\\log N}\\prod_{i=3}^k\\log_iN\\leq R(N)\\leq \\frac{1}{\\log 2}\\log_r N\\left(\\frac{N}{\\log N} \\prod_{i=3}^r \\log_iN\\right), $ valid for any $k\\geq 4$ with $\\log_kN\\geq k$ and any $r\\geq 1$ with $\\log_{2r}N\\geq 1$. (In these bounds $\\log_in$ denotes the $i$-fold iterated logarithm.)\nSee also [320].\nReferences\n\n\n[BlEr75] Bleicher, M. N. and Erdos, P., The number of distinct subsums of $\\sum \\sb{1}\\spN\\,1/i$. Math. Comp. (1975), 29-42.\n\n[BlEr76b] Bleicher, Michael N. and Erdos, Paul, Denominators of Egyptian fractions. II. Illinois J. Math. (1976), 598-613.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2048, "problem_number": "EP-322", "title": "Erdős Problem #322", "statement": "Let $k\\geq 3$ and $A\\subset \\mathbb{N}$ be the set of $k$th powers. What is the order of growth of $1_A^{(k)}(n)$, i.e. the number of representations of $n$ as the sum of $k$ many $k$th powers? Does there exist some $c>0$ and infinitely many $n$ such that $ 1_A^{(k)}(n) >n^c? $ ", "background": "Connected to Waring's problem. The famous Hypothesis $K$ of Hardy and Littlewood was that $1_A^{(k)}(n)\\leq n^{o(1)}$, but this was disproved by Mahler \\cite{Ma36} for $k=3$, who constructed infinitely many $n$ such that $ 1_A^{(3)}(n)\\gg n^{1/12} $ (where $A$ is the set of cubes). Erdos believed Hypothesis $K$ fails for all $k\\geq 4$, but this is unknown. Hardy and Littlewood made the weaker Hypothesis $K^*$ that for all $N$ and $\\epsilon>0$ $ \\sum_{n\\leq N}1_A^{(k)}(n)^2 \\ll_\\epsilon N^{1+\\epsilon}. $ Erdos and Graham remark: 'This is probably true but no doubt very deep. However, it would suffice for most applications.'\nIndependently Erdos \\cite{Er36} and Chowla proved that for all $k\\geq 3$ and infinitely many $n$ $ 1_A^{(k)}(n) \\gg n^{c/\\log\\log n} $ for some constant $c>0$ (depending on $k$). In \\cite{Er65b} Erdos claims an unpublished proof that, if $B$ is the set of $k$th powers of any set of positive density, then $ \\limsup 1_B^{(k)}(n)=\\infty. $ This is discussed in problem D4 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Er36] Erd\"{o}s, Paul, On the Representation of an Integer as the Sum of k k-th\nPowers. J. London Math. Soc. (1936), 133-136.\n\n[Er65b] Erdos, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Ma36] Mahler, Kurt, Note on Hypothesis K of Hardy and Littlewood. J. London Math. Soc. (1936), 136-138.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2049, "problem_number": "EP-323", "title": "Erdős Problem #323", "statement": "Let $1\\leq m\\leq k$ and $f_{k,m}(x)$ denote the number of integers $\\leq x$ which are the sum of $m$ many nonnegative $k$th powers. Is it true that $ f_{k,k}(x) \\gg_\\epsilon x^{1-\\epsilon} $ for all $\\epsilon>0$? Is it true that if $m0$.\nFor $k>2$ it is not known if $f_{k,k}(x)=o(x)$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2050, "problem_number": "EP-324", "title": "Erdős Problem #324", "statement": "Does there exist a polynomial $f(x)\\in\\mathbb{Z}[x]$ such that all the sums $f(a)+f(b)$ with $af(n)$.)", "background": "Erdos originally asked if even $f(n)\\leq n^{1/3}$ is true. This is known, and the best bound is due to Balog \\cite{Ba89} who proved that $ f(n) \\ll_\\epsilon n^{\\frac{4}{9\\sqrt{e}}+\\epsilon} $ for all $\\epsilon>0$. (Note $\\frac{4}{9\\sqrt{e}}=0.2695\\ldots$.)\nIt is likely that $f(n)\\leq n^{o(1)}$, or even $f(n)\\leq e^{O(\\sqrt{\\log n})}$.\nSee also Problem 59 on Green's open problems list.\nReferences\n\n\n[Ba89] Balog, A., On additive representation of integers. Acta Math. Hungar. (1989), 297-301.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2058, "problem_number": "EP-335", "title": "Erdős Problem #335", "statement": "Let $d(A)$ denote the density of $A\\subseteq \\mathbb{N}$. Characterise those $A,B\\subseteq \\mathbb{N}$ with positive density such that $ d(A+B)=d(A)+d(B). $ ", "background": "One way this can happen is if there exists $\\theta>0$ such that $ A=\\{ n>0 : \\{ n\\theta\\} \\in X_A\\}\\textrm{ and }B=\\{ n>0 : \\{n\\theta\\} \\in X_B\\} $ where $\\{x\\}$ denotes the fractional part of $x$ and $X_A,X_B\\subseteq \\mathbb{R}/\\mathbb{Z}$ are such that $\\mu(X_A+X_B)=\\mu(X_A)+\\mu(X_B)$. Are all possible $A$ and $B$ generated in a similar way (using other groups)?\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2059, "problem_number": "EP-336", "title": "Erdős Problem #336", "statement": "For $r\\geq 2$ let $h(r)$ be the maximal finite $k$ such that there exists a basis $A\\subseteq \\mathbb{N}$ of order $r$ (so every large integer is the sum of at most $r$ integers from $A$) and exact order $k$ (so every large integer is the sum of exactly $k$ integers from $A$).\nFind the value of $ \\lim_r \\frac{h(r)}{r^2}. $ ", "background": "A simple example of the order of a basis differing from the exact order is given by $A=\\cup_{k\\geq 0}(2^{2k},2^{2k+1}]$, which has order $2$ but exact order $3$.\nErdos and Graham \\cite{ErGr80b} have shown that a basis $A$ has an exact order if and only if $a_2-a_1,a_3-a_2,a_4-a_3,\\ldots$ are coprime. They also proved that $ \\frac{1}{4}\\leq \\lim_r \\frac{h(r)}{r^2}\\leq \\frac{5}{4}. $ The best bounds known for the limit are $ \\frac{1}{3}\\leq \\lim_r \\frac{h(r)}{r^2}\\leq \\frac{1}{2}, $ the lower bound originally due to Grekos \\cite{Gr88} and the upper bound to Nash \\cite{Na93}. Improved bounds in the lower order terms were given by Plagne \\cite{Pl04}.\nErdos and Graham \\cite{ErGr80b} showed $h(2)=4$. Nash \\cite{Na93} showed $h(3)=7$. The value of $h(4)$ is unknown, but it is known \\cite{Pl04} that $10\\leq h(4)\\leq 11$.\nReferences\n\n\n[ErGr80b] Erdos, P. and Graham, R. L., On bases with an exact order. Acta Arith. (1980), 201-207.\n\n[Gr88] Grekos, Georges, Sur l'ordre d'une base additive. ([1988?]), Exp. No. 31, 13.\n\n[Na93] Nash, John C. M., Some applications of a theorem of {M}. {K}neser. J. Number Theory (1993), 1--8.\n\n[Pl04] Plagne, Alain, \\`A{} propos de la fonction {$X$} d'{E}rd\\H{o}s et {G}raham. Ann. Inst. Fourier (Grenoble) (2004), 1717--1767.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2060, "problem_number": "EP-338", "title": "Erdős Problem #338", "statement": "The restricted order of a basis is the least integer $t$ (if it exists) such that every large integer is the sum of at most $t$ distinct summands from $A$. What are necessary and sufficient conditions that this exists? Can it be bounded (when it exists) in terms of the order of the basis? What are necessary and sufficient conditions that this is equal to the order of the basis?", "background": "Bateman has observed that for $h\\geq 3$ there is a basis of order $h$ with no restricted order, taking $ A=\\{1\\}\\cup \\{x>0 : h\\mid x\\}. $ Kelly \\cite{Ke57} has shown that any basis of order $2$ has restricted order at most $4$ and conjectured it always has restricted order at most $3$ (which he proved under the additional assumption that the basis has positive lower density). Kelly's conjecture was disproved by Hennecart \\cite{He05}, who constructed a basis of order $2$ with restricted order $4$.\nThe set of squares has order $4$ and restricted order $5$ (see \\cite{Pa33}) and the set of triangular numbers has order $3$ and restricted order $3$ (see \\cite{Sc54}).\nIs it true that if $A\\backslash F$ is a basis for all finite sets $F$ then $A$ must have a restricted order? What if they are all bases of the same order?\nHegyv\\'{a}ri, Hennecart, and Plagne \\cite{HHP07} have shown that for all $k\\geq2$ there exists a basis of order $k$ which has restricted order at least $ 2^{k-2}+k-1. $ \nReferences\n\n\n[HHP07] Hegyv\\'ari, Norbert and Hennecart, Fran\\c cois and Plagne,\nAlain, Answer to a question by {B}urr and {E}rd\\H{o}s on restricted\naddition, and related results. Combin. Probab. Comput. (2007), 747--756.\n\n[He05] Hennecart, Fran\\c cois, On the restricted order of asymptotic bases of order two. Ramanujan J. (2005), 123--130.\n\n[Ke57] Kelly, John B., Restricted bases. Amer. J. Math. (1957), 258-264.\n\n[Pa33] Pall, Gordon, On Sums of Squares. Amer. Math. Monthly (1933), 10-18.\n\n[Sc54] Schinzel, A., Sur la d\\'{e}composition des nombres naturels en sommes de nombres triangulaires distincts. Bull. Acad. Polon. Sci. Cl. III. (1954), 409-410.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2061, "problem_number": "EP-340", "title": "Erdős Problem #340", "statement": "Let $A=\\{1,2,4,8,13,21,31,45,66,81,97,\\ldots\\}$ be the greedy Sidon sequence: we begin with $1$ and iteratively include the next smallest integer that preserves the Sidon property (i.e. there are no non-trivial solutions to $a+b=c+d$). What is the order of growth of $A$? Is it true that $ \\lvert A\\cap \\{1,\\ldots,N\\}\\rvert \\gg N^{1/2-\\epsilon} $ for all $\\epsilon>0$ and large $N$?", "background": "This sequence is sometimes called the Mian-Chowla sequence. It is trivial that this sequence grows at least like $\\gg N^{1/3}$.\nErdos and Graham \\cite{ErGr80} also asked about the difference set $A-A$, whether this has positive density, and whether this contains $22$. It does contain $22$, since $a_{15}-a_{14}=204-182=22$. The smallest integer which is unknown to be in $A-A$ is $33$ (see A080200). It may be true that all or almost all integers are in $A-A$.\nThis sequence is at OEIS A005282.\nSee also [156].\nReferences\n\n\n[ErGr80] Erdos, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2062, "problem_number": "EP-341", "title": "Erdős Problem #341", "statement": "Let $A=\\{a_1<\\cdotsa_n$ which can be expressed uniquely as $a_i+a_j$ for $iT(n^{k+1})$?", "background": "Erdos and Graham \\cite{ErGr80} remark that very little is known about $T(A)$ in general. It is known that $ T(n)=1, T(n^2)=128, T(n^3)=12758, $ $ T(n^4)=5134240,\\textrm{ and }T(n^5)=67898771. $ Erdos and Graham remark that a good candidate for the $n$ in the question are $k=2^t$ for large $t$, perhaps even $t=3$, because of the highly restricted values of $n^{2^t}$ modulo $2^{t+1}$.\nReferences\n\n\n[ErGr80] Erdos, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2065, "problem_number": "EP-346", "title": "Erdős Problem #346", "statement": "Let $A=\\{1\\leq a_1< a_2<\\cdots\\}$ be a set of integers such that\n{UL}\n{LI} $A\\backslash B$ is complete for any finite subset $B$ and {/LI}\n{LI} $A\\backslash B$ is not complete for any infinite subset $B$.{/LI}\n{/UL}\n(Here 'complete' means all sufficiently large integers can be written as a sum of distinct members of the sequence.)\nIs it true that if $a_{n+1}/a_n \\geq 1+\\epsilon$ for some $\\epsilon>0$ and all $n$ then $ \\lim_n \\frac{a_{n+1}}{a_n}=\\frac{1+\\sqrt{5}}{2}? $ ", "background": "Graham \\cite{Gr64d} has shown that the sequence $a_n=F_n-(-1)^{n}$, where $F_n$ is the $n$th Fibonacci number, has these properties. Erdos and Graham \\cite{ErGr80} remark that it is easy to see that if $a_{n+1}/a_n>\\frac{1+\\sqrt{5}}{2}$ then the second property is automatically satisfied, and that it is not hard to construct very irregular sequences satisfying both properties.\nReferences\n\n\n[ErGr80] Erdos, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).\n\n[Gr64d] Graham, R. L., A property of Fibonacci numbers. Fibonacci Quart. (1964), 1-10.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2066, "problem_number": "EP-348", "title": "Erdős Problem #348", "statement": "For what values of $0\\leq m0$ and all $1<\\alpha < \\frac{1+\\sqrt{5}}{2}$. Proving this seems very difficult, since we do not even know whether $\\lfloor (3/2)^n\\rfloor$ is odd or even infinitely often.\nReferences\n\n\n[Gr64e] Graham, R. L., On a conjecture of Erdos in additive number theory. Acta Arith. (1964/65), 63-70.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2068, "problem_number": "EP-351", "title": "Erdős Problem #351", "statement": "Let $p(x)\\in \\mathbb{Q}[x]$. Is it true that $ A=\\{ p(n)+1/n : n\\in \\mathbb{N}\\} $ is strongly complete, in the sense that, for any finite set $B$, $ \\left\\{\\sum_{n\\in X}n : X\\subseteq A\\backslash B\\textrm{ finite }\\right\\} $ contains all sufficiently large integers?", "background": "Graham \\cite{Gr63} proved this is true when $p(n)=n$. Erdos and Graham also ask which rational functions $r(x)\\in\\mathbb{Z}(x)$ force $\\{ r(n) : n\\in\\mathbb{N}\\}$ to be complete?\nGraham \\cite{Gr64f} gave a complete characterisation of which polynomials $r\\in \\mathbb{R}[x]$ are such that $\\{ r(n) : n\\in \\mathbb{N}\\}$ is complete.\nIn the comments van Doorn has noted that a positive solution for $p(n)=n^2$ follows from \\cite{Gr63} together with result of Alekseyev \\cite{Al19} mentioned in [283].\nReferences\n\n\n[Al19] Alekseyev, Max A., On partitions into squares of distinct integers whose\nreciprocals sum to 1. (2019), 213--221.\n\n[Gr63] Graham, R. L., A theorem on partitions. J. Austral. Math. Soc. (1963), 435-441.\n\n[Gr64f] Graham, R. L., Complete sequences of polynomial values. Duke Math. J. (1964), 275-285.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2069, "problem_number": "EP-352", "title": "Erdős Problem #352", "statement": "Is there some $c>0$ such that every measurable $A\\subseteq \\mathbb{R}^2$ of measure $\\geq c$ contains the vertices of a triangle of area 1?", "background": "Erdos (unpublished) proved that this is true if $A$ has infinite measure, or if $A$ is an unbounded set of positive measure (stating in \\cite{Er78d} and \\cite{Er83d} it 'follows easily from the Lebesgue density theorem').\nIn \\cite{Er78d} and \\cite{Er83d} he speculated that perhaps $C=4\\pi/\\sqrt{27}\\approx 2.418$ works, which would be the best possible, as witnessed by a circle of radius $<2\\cdot 3^{-3/4}$.\nFurther evidence for this is given by a result of Freiling and Mauldin \\cite{Ma02}, who proved that if $A$ has outer measure $>4\\pi/\\sqrt{27}$ then $A$ contains the vertices of a triangle with area $>1$. This also proves the same threshold for the original problem under the assumption that $A$ is a compact convex set.\nMauldin also discusses this problem in \\cite{Ma13}, in which he notes that it suffices to prove this under the assumption that $A$ is the union of the interiors of $n<\\infty$ many compact convex sets. Freiling and Mauldin (see \\cite{Ma13}) have proved this conjecture if $1\\leq n\\leq 3$.\nReferences\n\n\n[Er78d] Erdos, P., Set-theoretic, measure-theoretic, combinatorial, and\nnumber-theoretic problems concerning point sets in Euclidean\nspace. Real Anal. Exchange (1978/79), 113-138.\n\n[Er83d] Erdos, Paul, Some combinatorial, geometric and set theoretic problems in measure theory. Measure Theory, Oberwolfach 1983: Proceedings of the Conference held at Oberwolfach, June 26-July 2, 1983 (1984), 321-327.\n\n[Ma02] Mauldin, R. D., Some problems in set theory, analysis and geometry. (2002), 493--506.\n\n[Ma13] Mauldin, R. Daniel, Some problems and ideas of {E}rd\\H{o}s in analysis and\ngeometry. (2013), 365--376.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2070, "problem_number": "EP-354", "title": "Erdős Problem #354", "statement": "Let $\\alpha,\\beta\\in \\mathbb{R}_{>0}$ such that $\\alpha/\\beta$ is irrational. Is the multiset $ \\{ \\lfloor \\alpha\\rfloor,\\lfloor 2\\alpha\\rfloor,\\lfloor 4\\alpha\\rfloor,\\ldots\\}\\cup \\{ \\lfloor \\beta\\rfloor,\\lfloor 2\\beta\\rfloor,\\lfloor 4\\beta\\rfloor,\\ldots\\} $ complete? That is, can all sufficiently large natural numbers $n$ be written as $ n=\\sum_{s\\in S}\\lfloor 2^s\\alpha\\rfloor+\\sum_{t\\in T}\\lfloor 2^t\\beta\\rfloor $ for some finite $S,T\\subset \\mathbb{N}$?\nWhat if $2$ is replaced by some $\\gamma\\in(1,2)$?", "background": "This question was first mentioned by Graham \\cite{Gr71}.\nHegyv\\'{a}ri \\cite{He89} proved that this holds if $\\alpha=m/2^n$ is a dyadic rational and $\\beta$ is not. He later \\cite{He91} proved that, for any fixed $\\alpha>0$, the set of $\\beta$ for which this holds either has measure $0$ or infinite measure. In \\cite{He94} he proved that the set of $(\\alpha,\\beta)$ for which the corresponding set of sums does not contain an infinite arithmetic progression has cardinality continuum.\nHegyv\\'{a}ri \\cite{He89} proved that the sequence is not complete if $\\alpha\\geq 2$ and $\\beta =2^k\\alpha$ for some $k\\geq 0$. Jiang and Ma \\cite{JiMa24} and Fang and He \\cite{FaHe25} prove that the sequence is not complete if $1<\\alpha<2$ and $\\beta=2^k\\alpha$ for some sufficiently large $k$.\nIt is likely (and Hegyv\\'{a}ri conjectures) that the assumption $\\alpha/\\beta$ irrational can be weakened to $\\alpha/\\beta \neq 2^k$ and either $\\alpha$ or $\\beta$ not a dyadic rational.\nIn the comments van Doorn proves the sequence is complete if $\\alpha < 2<\\beta<3$, and also proves that if either $\\alpha$ or $\\beta$ is not a dyadic rational then the corresponding sequence with ceiling functions replacing the floor functions is complete.\nReferences\n\n\n[FaHe25] Fang, J.-H. and He, J.-Y., On a problem of {E}rd\\H{o}s and {G}raham. Acta Math. Hungar. (2025), 532--542.\n\n[Gr71] Graham, R. L., On sums of integers taken from a fixed sequence. (1971), 22--40.\n\n[He89] Hegyv\\'ari, N., Some remarks on a problem of {E}rd\\H{o}s and {G}raham. Acta Math. Hungar. (1989), 149--154.\n\n[He91] Hegyv\\'ari, N., On complete sequences. Ann. Univ. Sci. Budapest. E\"otv\"os Sect. Math. (1991), 7--10.\n\n[He94] Hegyv\\'ari, Norbert, On sumset of certain sets. Publ. Math. Debrecen (1994), 115--122.\n\n[JiMa24] Jiang, Xing-Wang and Ma, Wu-Xia, A conjecture of {H}egyv\\'ari. Int. J. Number Theory (2024), 915--933.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2071, "problem_number": "EP-357", "title": "Erdős Problem #357", "statement": "Let $1\\leq a_1<\\cdots 0$?", "background": "A problem of MacMahon, studied by Andrews \\cite{An75}. When $n=1$ this sequence begins $ 1,2,4,5,8,10,14,15,\\ldots. $ This sequence is A002048 in the OEIS. Andrews conjectures $ a_k\\sim \\frac{k\\log k}{\\log\\log k}. $ Porubsky \\cite{Po77} proved that, for any $\\epsilon>0$, there are infinitely many $k$ such that $ a_k < (\\log k)^\\epsilon \\frac{k\\log k}{\\log\\log k}, $ and also that if $A(x)$ counts the number of $a_i\\leq x$ then $ \\limsup \\frac{A(x)}{\\pi(x)}\\geq \\frac{1}{\\log 2} $ where $\\pi(x)$ counts the number of primes $\\leq x$.\nSee also [839].\nReferences\n\n\n[An75] Andrews, George E., Research Problems: Mac Mahon's Prime Numbers of Measurement. Amer. Math. Monthly (1975), 922-923.\n\n[Po77] Porubsk\\'y, \\v S., On {M}ac{M}ahon's segmented numbers and related sequences. Nieuw Arch. Wisk. (3) (1977), 403--408.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2074, "problem_number": "EP-361", "title": "Erdős Problem #361", "statement": "Let $c>0$ and $n$ be some large integer. What is the size of the largest $A\\subseteq \\{1,\\ldots,\\lfloor cn\\rfloor\\}$ such that $n$ is not a sum of a subset of $A$? Does this depend on $n$ in an irregular way?\n\",\n \"difficulty\": \"L1\"\n},{", "background": "", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2075, "problem_number": "EP-365", "title": "Erdős Problem #365", "statement": "Do all pairs of consecutive powerful numbers $n$ and $n+1$ come from solutions to Pell equations? In other words, must either $n$ or $n+1$ be a square?\nIs the number of such $n\\leq x$ bounded by $(\\log x)^{O(1)}$?", "background": "Erdos originally asked Mahler whether there are infinitely many pairs of consecutive powerful numbers, but Mahler immediately observed that the answer is yes from the infinitely many solutions to the Pell equation $x^2=2^3y^2+1$.\nThe list of $n$ such that $n$ and $n+1$ are both powerful is A060355 in the OEIS.\nThe answer to the first question is no: Golomb \\cite{Go70} observed that both $12167=23^3$ and $12168=2^33^213^2$ are powerful. Walker \\cite{Wa76} proved that the equation $ 7^3x^2=3^3y^2+1 $ has infinitely many solutions, giving infinitely many counterexamples.\nSee also [364].\nThis is discussed in problem B16 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Go70] Golomb, S. W., Powerful numbers. Amer. Math. Monthly (1970), 848-855.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Wa76] Walker, David T., Consecutive integer pairs of powerful numbers and related\nDiophantine equations. Fibonacci Quart. (1976), 111-116.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2076, "problem_number": "EP-367", "title": "Erdős Problem #367", "statement": "Let $B_2(n)$ be the 2-full part of $n$ (that is, $B_2(n)=n/n'$ where $n'$ is the product of all primes that divide $n$ exactly once). Is it true that, for every fixed $k\\geq 1$, $ \\prod_{n\\leq m0$, $ \\limsup \\frac{\\prod_{n\\leq m0$, there are infinitely many $n$ such that $F(n) <(\\log n)^{2+\\epsilon}$.\nPasten \\cite{Pa24b} has proved that $ F(n) \\gg \\frac{(\\log\\log n)^2}{\\log\\log\\log n}. $ The largest prime factors of $n(n+1)$ are listed as A074399 in the OEIS.\nReferences\n\n\n[Er76d] Erdos, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44.\n\n[Ma35] Mahler, Kurt, \"{U}ber den gr\"{o}ssten Primteiler spezieller Polynome zweiten Grades. Archiv f\"{u}r math. og naturvid (1935).\n\n[Pa24b] Pasten, Hector, The largest prime factor of {$n^2+1$} and improvements on\nsubexponential {$ABC$}. Invent. Math. (2024), 373--385.\n\n[Po18] P\\'{o}lya, Georg, Zur arithmetischen {U}ntersuchung der {P}olynome. Math. Z. (1918), 143--148.\n\n[Sc67b] Schinzel, A., On two theorems of Gelfond and some of their applications. Acta Arith. (1967/68), 177-236.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2078, "problem_number": "EP-369", "title": "Erdős Problem #369", "statement": "Let $\\epsilon>0$ and $k\\geq 2$. Is it true that, for all sufficiently large $n$, there is a sequence of $k$ consecutive integers in $\\{1,\\ldots,n\\}$ all of which are $n^\\epsilon$-smooth?", "background": "Erdos and Graham state that this is open even for $k=2$ and 'the answer should be affirmative but the problem seems very hard'.\nUnfortunately the problem is trivially true as written (simply taking $\\{1,\\ldots,k\\}$ and $n>k^{1/\\epsilon}$). There are (at least) two possible variants which are non-trivial, and it is not clear which Erdos and Graham meant. Let $P$ be the sequence of $k$ consecutive integers sought for. The potential strengthenings which make this non-trivial are:\n{UL}\n{LI}Each $m\\in P$ must be $m^\\epsilon$-smooth. If this is the problem then the answer is yes, which follows from a result of Balog and Wooley \\cite{BaWo98}: for any $\\epsilon>0$ and $k\\geq 2$ there exist infinitely many $m$ such that $m+1,\\ldots,m+k$ are all $m^\\epsilon$-smooth.{/LI}\n{LI}Each $m\\in P$ must be in $[n/2,n]$ (say). In this case a positive answer also follows from the result of Balog and Wooley \\cite{BaWo98} for infinitely many $n$, but the case of all sufficiently large $n$ is open.{/LI}\n{/UL}\nSee also [370] and [928].\nReferences\n\n\n[BaWo98] Balog, Antal and Wooley, Trevor D., On strings of consecutive integers with no large prime factors. J. Austral. Math. Soc. Ser. A (1998), 266-276.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2079, "problem_number": "EP-371", "title": "Erdős Problem #371", "statement": "Let $P(n)$ denote the largest prime factor of $n$. Show that the set of $n$ with $P(n) (0.2017-o(1))x, $ and the same lower bound for the complement.\nIn \\cite{Er79e} Erdos also asks whether, for every $\\alpha$, the density of the set of $n$ where $ P(n+1)>P(n)n^\\alpha $ exists.\nTer\"{a}v\"{a}inen \\cite{Te18} has proved that the logarithmic density of the set of $n$ for which $P(n)P(n)n^\\alpha$ exists and is equal to $ \\int_{[0,1]^2}1_{y\\geq x+\\alpha}u(x)u(y)\\mathrm{d}x\\mathrm{d}y $ where $u(x)=x^{-1}\\rho(x^{-1}-1)$ and $\\rho$ is the Dickman function. Wang \\cite{Wa21} has proved the same value holds for the asymptotic density (and in particular provided an affirmative answer to the original question) conditional on the Elliott-Halberstam conjecture for friable integers.\nThe sequence of such $n$ is A070089 in the OEIS.\nSee also [372] and [928].\nReferences\n\n\n[Er79e] Erdos, Paul, Some unconventional problems in number theory. Ast\\'{e}risque (1979), 73-82.\n\n[ErPo78] Erdos, Paul and Pomerance, Carl, On the largest prime factors of {$n$} and {$n+1$}. Aequationes Math. (1978), 311-321.\n\n[LuWa25] L\"u, Xiaodong and Wang, Zhiwei, On the largest prime factors of consecutive integers. Monatsh. Math. (2025), 403--418.\n\n[TaTe19] Tao, Terence and Ter\"{a}v\"{a}inen, Joni, The structure of correlations of multiplicative functions at\nalmost all scales, with applications to the {C}howla and\n{E}lliott conjectures. Algebra Number Theory (2019), 2103--2150.\n\n[Te18] Ter\"{a}v\"{a}inen, Joni, On binary correlations of multiplicative functions. Forum Math. Sigma (2018), Paper No. e10, 41.\n\n[Wa21] Wang, Zhiwei, Three conjectures on {$P^+(n)$} and {$P^+(n+1)$} hold under\nthe {E}lliott-{H}alberstam conjecture for friable integers. J. Number Theory (2021), 1--11.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2080, "problem_number": "EP-373", "title": "Erdős Problem #373", "statement": "Show that the equation $ n! = a_1!a_2!\\cdots a_k!, $ with $n-1>a_1\\geq a_2\\geq \\cdots \\geq a_k\\geq 2$, has only finitely many solutions.", "background": "This would follow if $P(n(n+1))/\\log n\\to \\infty$, where $P(m)$ denotes the largest prime factor of $m$ (see Problem [368]). Erdos \\cite{Er76d} proved that this problem would also follow from showing that $P(n(n-1))>4\\log n$.\nThe condition $a_1a_1\\geq a_2$ then $ a_1\\geq n-5\\log\\log n, $ and says it 'would be nice' to prove $a_1\\geq n-o(\\log\\log n)$. Bhat and Ramachandra \\cite{BhRa10} replace the $5$ with $(1+o(1))\\frac{1}{\\log 2}$, and also prove that the same bound holds for arbitrary $k\\geq 2$.\nNumerical investigations on solutions to $n!=a_1!a_2!$ have been carried out by Caldwell \\cite{Ca94} and Habsieger \\cite{Ha}, and it is known that there are no solutions aside from $10!=6!7!$ for $n\\leq 10^{3000}$.\nReferences\n\n\n[BhRa10] Bhat, K. Dzh. and Ramachandra, K., A remark on factorials that are products of factorials. Mat. Zametki (2010), 350--354.\n\n[Ca94] C. Caldwell, The Diophantine equation $A!B!=C!$. J. Recreat. Math. (1994), 128-133.\n\n[Er76d] Erdos, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44.\n\n[Er93] Erdos, Paul, Some of my favorite solved and unsolved problems in graph\ntheory. Quaestiones Math. (1993), 333-350.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Ha] Haight, J. A., Metric Diophantine approximation and related topics. PhD thesis ().\n\n[Lu07b] Luca, Florian, On factorials which are products of factorials. Math. Proc. Cambridge Philos. Soc. (2007), 533--542.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2081, "problem_number": "EP-374", "title": "Erdős Problem #374", "statement": "For any $m\\in \\mathbb{N}$, let $F(m)$ be the minimal $k\\geq 2$ (if it exists) such that there are $a_1<\\cdots 1\\}$,{/LI}\n{LI} $\\lvert D_3\\cap \\{1,\\ldots,n\\}\\rvert = o(\\lvert D_4\\cap \\{1,\\ldots,n\\}\\rvert)$,{/LI}\n{LI} the least element of $D_6$ is $527$, and{/LI}\n{LI} $D_k=\\emptyset$ for $k>6$.{/LI}\n{/UL}\nReferences\n\n\n[ErGr76] Erdos, P. and Graham, R. L., On products of factorials. Bull. Inst. Math. Acad. Sinica (1976), 337-355.\n\n[LSS14] Luca, F. and Saradha, N. and Shorey, T. N., Squares and factorials in products of factorials. Monatsh. Math. (2014), 385-400.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2082, "problem_number": "EP-376", "title": "Erdős Problem #376", "statement": "Are there infinitely many $n$ such that $\\binom{2n}{n}$ is coprime to $105$?", "background": "Erdos, Graham, Ruzsa, and Straus \\cite{EGRS75} have shown that, for any two odd primes $p$ and $q$, there are infinitely many $n$ such that $\\binom{2n}{n}$ is coprime to $pq$.\nThis is equivalent (via Kummer's theorem) to whether there are infinitely many $n$ which have only digits $0,1$ in base $3$, digits $0,1,2$ in base $5$, and digits $0,1,2,3$ in base $7$.\nThe sequence of such $n$ is A030979 in the OEIS.\nThe best result in this direction is due to Bloom and Croot \\cite{BlCr25}, who proved that, if $p_1,p_2,p_3$ are sufficiently large primes, then there are infinitely many $n$ such that almost all of the base $p_i$ digits are $0$, there are infinitely many $n$ such that $\\binom{2n}{n}$ is coprime to $p_1p_2p_3$, except for a factor of size $\\leq n^\\epsilon$.\nThis is mentioned in problem B33 of Guy's collection \\cite{Gu04}. It is also discussed in an article of Pomerance \\cite{Po15c}.\nGraham offered \\$1000 for a solution to this problem (as mentioned in \\cite{Gu04} and \\cite{BeHa98}).\nReferences\n\n\n[BeHa98] Berend, Daniel and Harmse, J\\o rgen E., On some arithmetical properties of middle binomial\ncoefficients. Acta Arith. (1998), 31--41.\n\n[BlCr25] T. F. Bloom and E. Croot, Integers with small digits in multiple bases. arXiv:2509.02835 (2025).\n\n[EGRS75] Erdos, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., On the prime factors of $(\\sp{2n}\\sb{n})$. Math. Comp. (1975), 83-92.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Po15c] Pomerance, Carl, Divisors of the middle binomial coefficient. Amer. Math. Monthly (2015), 636--644.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2083, "problem_number": "EP-377", "title": "Erdős Problem #377", "statement": "Is there some absolute constant $C>0$ such that $ \\sum_{p\\leq n}1_{p\nmid \\binom{2n}{n}}\\frac{1}{p}\\leq C $ for all $n$ (where the summation is restricted to primes $p\\leq n$)?", "background": "A question of Erdos, Graham, Ruzsa, and Straus \\cite{EGRS75}, who proved that if $f(n)$ is the sum in question then $ \\lim_{x\\to \\infty}\\frac{1}{x}\\sum_{n\\leq x}f(n) = \\sum_{k=2}^\\infty \\frac{\\log k}{2^k}=\\gamma_0 $ and $ \\lim_{x\\to \\infty}\\frac{1}{x}\\sum_{n\\leq x}f(n)^2 = \\gamma_0^2, $ so that for almost all integers $f(m)=\\gamma_0+o(1)$. They also prove that, for all large $n$, $ f(n) \\leq c\\log\\log n $ for some constant $c<1$. (It is trivial from Mertens estimates that $f(n)\\leq (1+o(1))\\log\\log n$.)\nA positive answer would imply that $ \\sum_{p\\leq n}1_{p\\mid \\binom{2n}{n}}\\frac{1}{p}=(1-o(1))\\log\\log n, $ and Erdos, Graham, Ruzsa, and Straus say there is 'no doubt' this latter claim is true.\nThis is mentioned in problem B33 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[EGRS75] Erdos, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., On the prime factors of $(\\sp{2n}\\sb{n})$. Math. Comp. (1975), 83-92.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2084, "problem_number": "EP-380", "title": "Erdős Problem #380", "statement": "We call an interval $[u,v]$ 'bad' if the greatest prime factor of $\\prod_{u\\leq m\\leq v}m$ occurs with an exponent greater than $1$. Let $B(x)$ count the number of $n\\leq x$ which are contained in at least one bad interval. Is it true that $ B(x)\\sim \\#\\{ n\\leq x: P(n)^2\\mid n\\}, $ where $P(n)$ is the largest prime factor of $n$?", "background": "Erdos and Graham only knew that $B(x) > x^{1-o(1)}$. Similarly, we call an interval $[u,v]$ 'very bad' if $\\prod_{u\\leq m\\leq v}m$ is powerful. The number of integers $n\\leq x$ contained in at least one very bad interval should be $\\ll x^{1/2}$. In fact, it should be asymptotic to the number of powerful numbers $\\leq x$.\nWe have $ \\#\\{ n\\leq x: P(n)^2\\mid n\\}=\\frac{x}{\\exp((c+o(1))\\sqrt{\\log x\\log\\log x})} $ for some constant $c>0$.\nTao notes in the comments that if $[u,v]$ is bad then it cannot contain any primes, and hence certainly $v<2u$, and in general $v-u$ must be small (for example, assuming Cramer's conjecture, $v-u\\ll (\\log u)^2$).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2085, "problem_number": "EP-382", "title": "Erdős Problem #382", "statement": "Let $u\\leq v$ be such that the largest prime dividing $\\prod_{u\\leq m\\leq v}m$ appears with exponent at least $2$. Is it true that $v-u=v^{o(1)}$? Can $v-u$ be arbitrarily large?", "background": "Erdos and Graham report it follows from results of Ramachandra that $v-u\\leq v^{1/2+o(1)}$.\nCambie has observed that the first question boils down to some old conjectures on prime gaps.\nBy Cram\\'{er's conjecture}, for every $\\epsilon>0,$ for every $u$ sufficiently large there is a prime between $u$ and $u+u^\\epsilon$.\nThus for $u+u^\\epsilon0$. For any fixed $k$, there is therefore a positive 'probability' that there are $k$ consecutive integers around $q^2$ (for a prime $q$) all of whose prime divisors are bounded above by $q$, when $v-u\\geq k$. See [383] for a conjecture along these lines. A similar argument applies if we replace multiplicity $2$ with multiplicity $r$, for any fixed $r\\geq 2$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2086, "problem_number": "EP-383", "title": "Erdős Problem #383", "statement": "Is it true that for every $k$ there are infinitely many primes $p$ such that the largest prime divisor of $ \\prod_{0\\leq i\\leq k}(p^2+i) $ is $p$?", "background": "A positive answer to this would give an answer to the second part of [382]. Heuristically, the 'probability' that $n$ has no prime divisors $\\geq n^{1/2}$ is $1-\\log 2>0$, so standard heuristics predict the answer to this is yes.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2087, "problem_number": "EP-385", "title": "Erdős Problem #385", "statement": "Let $ F(n) = \\max_{\\substack{mn$ for all sufficiently large $n$? Does $F(n)-n\\to \\infty$ as $n\\to\\infty$?", "background": "A question of Erdos, Eggleton, and Selfridge, who write that 'plausible conjectures on primes' imply that $F(n)\\leq n$ for only finitely many $n$, and in fact it is possible that this quantity is always at least $n+(1-o(1))\\sqrt{n}$ (note that it is trivially $\\leq n+\\sqrt{n}$).\nTao has discussed this problem in a blog post.\nSarosh Adenwalla has observed that the first question is equivalent to [430]. Indeed, if $n$ is large and $a_i$ is the sequence defined in the latter problem, then [430] implies that there is a composite $a_j$ such that $a_j-p(a_j)>n$ and hence $F(n)>n$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2088, "problem_number": "EP-386", "title": "Erdős Problem #386", "statement": "Let $2\\leq k\\leq n-2$. Can $\\binom{n}{k}$ be the product of consecutive primes infinitely often? For example $ \\binom{21}{2}=2\\cdot 3\\cdot 5\\cdot 7. $ ", "background": "Erdos and Graham write that 'a proof that this cannot happen infinitely often for $\\binom{n}{2}$ seems hopeless; probably this can never happen for $\\binom{n}{k}$ if $3\\leq k\\leq n-3$.'\nWeisenberg has provided four easy examples that show Erdos and Graham were too optimistic here: $ \\binom{7}{3}=5\\cdot 7, $ $ \\binom{10}{4}= 2\\cdot 3\\cdot 5\\cdot 7, $ $ \\binom{14}{4} = 7\\cdot 11\\cdot 13, $ and $ \\binom{15}{6}=5\\cdot 7\\cdot 11\\cdot 13. $ The known values of $n$ for which $\\binom{n}{2}$ is the product of consecutive primes are $4,6,15,21,715$ (see A280992).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2089, "problem_number": "EP-387", "title": "Erdős Problem #387", "statement": "Is there an absolute constant $c>0$ such that, for all $1\\leq k< n$, the binomial coefficient $\\binom{n}{k}$ has a divisor in $(cn,n]$?", "background": "Erdos once conjectured that $\\binom{n}{k}$ must always have a divisor in $(n-k,n]$, but this was disproved by Schinzel and Erdos \\cite{Sc58}. A counterexample is given by $n=99215$ and $k=15$. Schinzel conjectured (see problem B34 of \\cite{Gu04}) that, for all sufficiently large $k$ which are not prime powers, there exists an $n$ such that $\\binom{n}{k}$ is not divisible by any integer in $(n-k,n]$.\nIt is easy to see that $\\binom{n}{k}$ always has a divisor in $[n/k,n]$.\nFaulkner \\cite{Fa66} proved that, if $p$ is the least prime $>2k$ and $n\\geq p$, then $\\binom{n}{k}$ has a prime divisor $\\geq p$ (except $\\binom{9}{2}$ and $\\binom{10}{3}$).\nThis is discussed in problems B33 and B34 of Guy's collection \\cite{Gu04}, who says that Erdos conjectured this is true for any $c<1$ (if $n$ is sufficiently large).\nReferences\n\n\n[Fa66] Faulkner, M., On a theorem of {S}ylvester and {S}chur. J. London Math. Soc. (1966), 107--110.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Sc58] Schinzel, A., Sur un probl\\`eme de {P}. {E}rd\\H{o}s. Colloq. Math. (1958), 198--204.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2090, "problem_number": "EP-388", "title": "Erdős Problem #388", "statement": "Can one classify all solutions of $ \\prod_{1\\leq i\\leq k_1}(m_1+i)=\\prod_{1\\leq j\\leq k_2}(m_2+j) $ where $k_1,k_2>3$ and $m_1+k_1\\leq m_2$? Are there only finitely many solutions?", "background": "More generally, if $k_1>2$ then for fixed $a$ and $b$ $ a\\prod_{1\\leq i\\leq k_1}(m_1+i)=b\\prod_{1\\leq j\\leq k_2}(m_2+j) $ should have only a finite number of solutions.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2091, "problem_number": "EP-389", "title": "Erdős Problem #389", "statement": "Is it true that for every $n\\geq 1$ there is a $k$ such that $ n(n+1)\\cdots(n+k-1)\\mid (n+k)\\cdots (n+2k-1)? $ ", "background": "Asked by Erdos and Straus.\nFor example when $n=2$ we have $k=5$: $ 2\\times 3 \\times 4 \\times 5\\times 6 \\mid 7 \\times 8 \\times 9\\times 10\\times 11. $ and when $n=3$ we have $k=4$: $ 3\\times 4\\times 5\\times 6 \\mid 7\\times 8\\times 9\\times 10. $ Bhavik Mehta has computed the minimal such $k$ for $1\\leq n\\leq 18$ (now available as A375071 on the OEIS).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2092, "problem_number": "EP-390", "title": "Erdős Problem #390", "statement": "Let $f(n)$ be the minimal $m$ such that $ n! = a_1\\cdots a_k $ with $n< a_1<\\cdots 0$?\nIs it true that, for $k\\geq 2$, $ \\sum_{n\\leq x}t_{k+1}(n) =o\\left(\\sum_{n\\leq x}t_k(n)\\right)? $ ", "background": "In \\cite{ErGr80} they mention a conjecture of Erdos that the sum is $o(x^2)$. This was proved by Erdos and Hall \\cite{ErHa78}, who proved that in fact $ \\sum_{n\\leq x}t_2(n)\\ll \\frac{\\log\\log\\log x}{\\log\\log x}x^2. $ Erdos and Hall conjecture that the sum is $o(x^2/(\\log x)^c)$ for any $c<\\log 2$.\nSince $t_2(p)=p-1$ for prime $p$ it is trivial that $ \\sum_{n\\leq x}t_2(n)\\gg \\frac{x^2}{\\log x}. $ Erdos and Hall \\cite{ErHa78} also note that $t_{n-1}(n!)=2$ and $t_{n-2}(n!)\\ll n$, which $n=2^r$ shows is the best possible. They ask about the behaviour of $t_{n-3}(n!)$ and also ask ask whether, for infinitely many $n$, $ t_k(n!)< t_{k-1}(n!)-1 $ for all $1\\leq k0$ such that there are infinitely many $n$ where $m+\\epsilon \\omega(m)\\leq n$ for all $m\\phi(m+2)>\\cdots \\phi(m+k)? $ Is it true that 'natural' ordering which mimics what happens to $\\phi(1),\\ldots,\\phi(k)$ is the most likely to appear?", "background": "Erdos \\cite{Er36b} proved that $ F(n)\\asymp \\log\\log\\log n, $ and similarly if we replace $\\phi$ with $\\sigma$ or $\\tau$ or $\nu$ or any 'decent' additive or multiplicative function.\nWeisenberg has observed that the same questions could be asked for ordering patterns which allow equality (indeed, the final problem only makes sense if we allow equality).\nReferences\n\n\n[Er36b] Erdos, P., On a problem of Chowla and some related problems. Proc. Cambridge Philos. Soc. (1936), 530-540.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2107, "problem_number": "EP-416", "title": "Erdős Problem #416", "statement": "Let $V(x)$ count the number of $n\\leq x$ such that $\\phi(m)=n$ is solvable. Does $V(2x)/V(x)\\to 2$? Is there an asymptotic formula for $V(x)$?", "background": "Pillai \\cite{Pi29} proved $V(x)=o(x)$. Erdos \\cite{Er35b} proved $V(x)=x(\\log x)^{-1+o(1)}$.\nThe behaviour of $V(x)$ is now almost completely understood. Maier and Pomerance \\cite{MaPo88} proved $ V(x)=\\frac{x}{\\log x}e^{(C+o(1))(\\log\\log\\log x)^2}, $ for some explicit constant $C>0$. Ford \\cite{Fo98} improved this to $ V(x)\\asymp\\frac{x}{\\log x}e^{C_1(\\log\\log\\log x-\\log\\log\\log\\log x)^2+C_2\\log\\log\\log x-C_3\\log\\log\\log\\log x} $ for some explicit constants $C_1,C_2,C_3>0$. Unfortunately this falls just short of an asymptotic formula for $V(x)$ and determining whether $V(2x)/V(x)\\to 2$.\nIn \\cite{Er79e} Erdos asks further to estimate the number of $n\\leq x$ such that the smallest solution to $\\phi(m)=n$ satisfies $kx1$?", "background": "It is trivial that $V'(x) \\leq V(x)$. In \\cite{Er98} Erdos suggests the limit may be infinite. See also [416].\nReferences\n\n\n[Er98] Erdos, Paul, Some of my new and almost new problems and results in combinatorial number theory. Number theory (Eger, 1996) (1998), 169-180.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2109, "problem_number": "EP-420", "title": "Erdős Problem #420", "statement": "If $\\tau(n)$ counts the number of divisors of $n$ then let $ F(f,n)=\\frac{\\tau((n+\\lfloor f(n)\\rfloor)!)}{\\tau(n!)}. $ Is it true that $ \\lim_{n\\to \\infty}F((\\log n)^C,n)=\\infty $ for large $C$?\nIs it true that $F(\\log n,n)$ is everywhere dense in $(1,\\infty)$?\nMore generally, if $f(n)\\leq \\log n$ is a monotonic function such that $f(n)\\to \\infty$ as $n\\to \\infty$, then is $F(f,n)$ everywhere dense?", "background": "Erdos and Graham write that it is easy to show that $\\lim F(n^{1/2},n)=\\infty$, and in fact the $n^{1/2}$ can be replaced by $n^{1/2-c}$ for some small constant $c>0$.\nErdos, Graham, Ivi\\'{c}, and Pomerance \\cite{EGIP96} have proved that $ \\liminf F(c\\log n, n) = 1 $ for any $c>0$, and $ \\lim F(n^{4/9},n)=\\infty. $ (The exponent $4/9$ can be improved slightly.) They also prove that, if $f(n)=o((\\log n)^2)$, then for almost all $n$ $ F(f,n)\\sim 1. $ van Doorn notes in the comments that the existence of infinitely many bounded prime gaps implies $ \\limsup_{n\\to \\infty}F(g(n),n)=\\infty $ for any $g(n)\\to \\infty$, and that Cram\\'{e}r's conjecture implies $ \\lim F(g(n)(\\log n)^2, n)=\\infty $ for any $g(n)\\to \\infty$>\nReferences\n\n\n[EGIP96] Erdos, Paul and Graham, S. W. and Ivi\\'c, Aleksandar and\nPomerance, Carl, On the number of divisors of {$n!$}. (1996), 337--355.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2110, "problem_number": "EP-421", "title": "Erdős Problem #421", "statement": "Is there a sequence $1\\leq d_11/e-\\epsilon$ for any $\\epsilon>0$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2111, "problem_number": "EP-422", "title": "Erdős Problem #422", "statement": "Let $f(1)=f(2)=1$ and for $n>2$ $ f(n) = f(n-f(n-1))+f(n-f(n-2)). $ Does $f(n)$ miss infinitely many integers? What is its behaviour?", "background": "Asked by Hofstadter. The sequence begins $1,1,2,3,3,4,\\ldots$ and is A005185 in the OEIS. It is not even known whether $f(n)$ is well-defined for all $n$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2112, "problem_number": "EP-423", "title": "Erdős Problem #423", "statement": "Let $a_1=1$ and $a_2=2$ and for $k\\geq 3$ choose $a_k$ to be the least integer $>a_{k-1}$ which is the sum of at least two consecutive terms of the sequence. What is the asymptotic behaviour of this sequence?", "background": "Asked by Hofstadter (in \\cite{Er77c} Erdos says Hofstadter was inspired by a similar question of Ulam). The sequence begins $ 1,2,3,5,6,8,10,11,\\ldots $ and is A005243 in the OEIS.\nBolan and Tang have independently proved that there are infinitely many integers which do not appear in this sequence. In fact, the sequence $a_n-n$ is nondecreasing and unbounded.\nReferences\n\n\n[Er77c] Erdos, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ.,\nNew York, 1976) (1977), 43-72.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2113, "problem_number": "EP-424", "title": "Erdős Problem #424", "statement": "Let $a_1=2$ and $a_2=3$ and continue the sequence by appending to $a_1,\\ldots,a_n$ all possible values of $a_ia_j-1$ with $i\neq j$. Is it true that the set of integers which eventually appear has positive density?", "background": "Asked by Hofstadter. The sequence begins $2,3,5,9,14,17,26,\\ldots$ and is A005244 in the OEIS. This problem is also discussed in section E31 of Guy's book Unsolved Problems in Number Theory.\nIn \\cite{ErGr80} (and in Guy's book) this problem as written is asking for whether almost all integers appear in this sequence, but the answer to this is trivially no (as pointed out to me by Steinerberger): no integer $\\equiv 1\\pmod{3}$ is ever in the sequence, so the set of integers which appear has density at most $2/3$. This is easily seen by induction, and the fact that if $a,b\\in \\{0,2\\}\\pmod{3}$ then $ab-1\\in \\{0,2\\}\\pmod{3}$.\nPresumably it is the weaker question of whether a positive density of integers appear (as correctly asked in \\cite{Er77c}) that was also intended in \\cite{ErGr80}.\nSee also Problem 63 of Green's open problems list.\nReferences\n\n\n[Er77c] Erdos, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ.,\nNew York, 1976) (1977), 43-72.\n\n[ErGr80] Erdos, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2114, "problem_number": "EP-425", "title": "Erdős Problem #425", "statement": "Let $F(n)$ be the maximum possible size of a subset $A\\subseteq\\{1,\\ldots,N\\}$ such that the products $ab$ are distinct for all $a0? $ ", "background": "Erdos and Graham could show this is true (assuming the prime $k$-tuple conjecture) if we replace $\\liminf$ by $\\limsup$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2116, "problem_number": "EP-430", "title": "Erdős Problem #430", "statement": "Fix some integer $n$ and define a decreasing sequence in $[1,n)$ by $a_1=n-1$ and, for $k\\geq 2$, letting $a_k$ be the greatest integer in $[1,a_{k-1})$ such that all of the prime factors of $a_k$ are $>n-a_k$.\nIs it true that, for sufficiently large $n$, not all of this sequence can be prime?", "background": "Erdos and Graham write 'preliminary calculations made by Selfridge indicate that this is the case but no proof is in sight'. For example if $n=8$ we have $a_1=7$ and $a_2=5$ and then must stop.\nSarosh Adenwalla has observed that this problem is equivalent to (the first part of) [385]. Indeed, assuming a positive answer to that, for all large $n$, there exists a composite $mn-m$. It follows that such an $m$ is equal to some $a_i$ in the sequence defined for $[1,n)$, and $m$ is composite by assumption.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2117, "problem_number": "EP-431", "title": "Erdős Problem #431", "statement": "Are there two infinite sets $A$ and $B$ such that $A+B$ agrees with the set of prime numbers up to finitely many exceptions?", "background": "A problem of Ostmann, sometimes known as the 'inverse Goldbach problem'. The answer is surely no. The best result in this direction is due to Elsholtz and Harper \\cite{ElHa15}, who showed that if $A,B$ are such sets then for all large $x$ we must have $ \\frac{x^{1/2}}{\\log x\\log\\log x} \\ll \\lvert A \\cap [1,x]\\rvert \\ll x^{1/2}\\log\\log x $ and similarly for $B$.\nElsholtz \\cite{El01} has proved there are no sets $A,B,C$ (all of size at least $2$) such that $A+B+C$ agrees with the set of prime numbers up to finitely many exceptions.\nGranville \\cite{Gr90} proved, conditional on the prime $k$-tuples conjecture, that there are infinite sets $B$ and $C$ such that $ \\{ \\tfrac{b+c}{2}: b\\in B, c\\in C\\} $ is a subset of the primes. Tao and Ziegler \\cite{TaZi23} gave an unconditional proof that there are infinite sets $B=\\{b_1<\\cdots\\}$ and $C=\\{c_1<\\cdots\\}$ such that $ \\{ b_i+c_j : b_i\\in B, c_j\\in C, i1/2$, if $p$ is a sufficiently large prime then, for any $n\\geq 0$, there exist $a,b\\in(n,n+p^c)$ such that $ab\\equiv 1\\pmod{p}$?", "background": "Heilbronn (unpublished) proved this for $c$ sufficiently close to $1$. Heath-Brown \\cite{He00} used Kloosterman sums to prove this for all $c>3/4$.\nThis is discussed in this MathOverflow question.\nReferences\n\n\n[He00] Heath-Brown, D. R., Arithmetic applications of {K}loosterman sums. Nieuw Arch. Wiskd. (5) (2000), 380--384.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2121, "problem_number": "EP-450", "title": "Erdős Problem #450", "statement": "How large must $y=y(\\epsilon,n)$ be such that the number of integers in $(x,x+y)$ with a divisor in $(n,2n)$ is at most $\\epsilon y$?", "background": "It is not clear what the intended quantifier on $x$ is. Cambie has observed that if this is intended to hold for all $x$ then, provided $ \\epsilon(\\log n)^\\delta (\\log\\log n)^{3/2}\\to \\infty $ as $n\\to \\infty$, where $\\delta=0.086\\cdots$, there is no such $y$, which follows from an averaging argument and the work of Ford \\cite{Fo08}.\nOn the other hand, Cambie has observed that if $\\epsilon\\ll 1/n$ then $y(\\epsilon,n)\\sim 2n$: indeed, if $y<2n$ then this is impossible taking $x+n$ to be a multiple of the lowest common multiple of $\\{n+1,\\ldots,2n-1\\}$. On the other hand, for every fixed $\\delta\\in (0,1)$ and $n$ large every $2(1+\\delta)n$ consecutive elements contains many elements which are a multiple of an element in $(n,2n)$.\nReferences\n\n\n[Fo08] Ford, Kevin, The distribution of integers with a divisor in a given\ninterval. Ann. of Math. (2) (2008), 367-433.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2122, "problem_number": "EP-451", "title": "Erdős Problem #451", "statement": "Estimate $n_k$, the smallest integer $>2k$ such that $\\prod_{1\\leq i\\leq k}(n_k-i)$ has no prime factor in $(k,2k)$.", "background": "Erdos and Graham write 'we can prove $n_k>k^{1+c}$ but no doubt much more is true'.\nIn \\cite{Er79d} Erdos writes that probably $n_kk^d$ for all constant $d$.\nAdenwalla observes that an easy upper bound is $n_k\\leq \\prod_{k\\log\\log n$ for all $n\\in I$?", "background": "Erdos \\cite{Er37} proved that the density of integers $n$ with $\\omega(n)>\\log\\log n$ is $1/2$. The Chinese remainder theorem implies that there is such an interval with $ \\lvert I\\rvert \\geq (1+o(1))\\frac{\\log x}{(\\log\\log x)^2}. $ It could be true that there is such an interval of length $(\\log x)^{k}$ for arbitrarily large $k$.\nReferences\n\n\n[Er37] Erd\"{o}s, Paul, Note on the number of prime divisors of integers. J. London Math. Soc. (1937), 308-314.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2124, "problem_number": "EP-454", "title": "Erdős Problem #454", "statement": "Let $ f(n) = \\min_{i0.352\\cdots. $ \nReferences\n\n\n[Ri76] Richter, Bernd, \"{U}ber die Monotonie von Differenzenfolgen. Acta Arith. (1976), 225-227.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2126, "problem_number": "EP-456", "title": "Erdős Problem #456", "statement": "Let $p_n$ be the smallest prime $\\equiv 1\\pmod{n}$ and let $m_n$ be the smallest integer such that $n\\mid \\phi(m_n)$.\nIs it true that $m_n0$ such that there are infinitely many $n$ where all primes $p\\leq (2+\\epsilon)\\log n$ divide $ \\prod_{1\\leq i\\leq \\log n}(n+i)? $ ", "background": "A problem of Erdos and Pomerance.\nMore generally, let $q(n,k)$ denote the least prime which does not divide $\\prod_{1\\leq i\\leq k}(n+i)$. This problem asks whether $q(n,\\log n)\\geq (2+\\epsilon)\\log n$ infinitely often. Taking $n$ to be the product of primes between $\\log n$ and $(2+o(1))\\log n$ gives an example where $ q(n,\\log n)\\geq (2+o(1))\\log n. $ Can one prove that $q(n,\\log n)<(1-\\epsilon)(\\log n)^2$ for all large $n$ and some $\\epsilon>0$?\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2128, "problem_number": "EP-460", "title": "Erdős Problem #460", "statement": "Let $a_0=0$ and $a_1=1$, and in general define $a_k$ to be the least integer $>a_{k-1}$ for which $(n-a_k,n-a_i)=1$ for all $0\\leq in$ (this observation was made by Svyable using ChatGPT).\nUnfortunately, although both sources mention a forthcoming paper of Eggleton, Erdos, and Selfridge, I cannot find a candidate paper of theirs with this problem in, and hence the motivation behind this problem, and what the precise problem intended was, is unclear.\nChojecki has noted in the comments that a positive solution to the main problem would follow if $ f(n) = \\sum_{aa}\\frac{1}{a}\\to \\infty, $ where $P^-(\\cdot)$ is the least prime factor. Standard estimates on rough numbers show that $\\frac{1}{N}\\sum_{n\\leq N}f(n)\\gg \\log\\log N$, so $f(n)$ does diverge on average, but it is unclear whether $f(n)\\to \\infty$ for all $n$.\nReferences\n\n\n[Er77c] Erdos, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ.,\nNew York, 1976) (1977), 43-72.\n\n[ErGr80] Erdos, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2129, "problem_number": "EP-461", "title": "Erdős Problem #461", "statement": "Let $s_t(n)$ be the $t$-smooth component of $n$ - that is, the product of all primes $p$ (with multiplicity) dividing $n$ such that $p0$ such that $ \\sum_{\\substack{n0$ such that $ \\sum_{x\\leq n\\leq x+Cx^{1/2}(\\log x)^2}\\frac{p(n)}{n} \\gg 1 $ for all large $x$?\n\",\n \"difficulty\": \"L1\"\n},{", "background": "", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2131, "problem_number": "EP-463", "title": "Erdős Problem #463", "statement": "Is there a function $f$ with $f(n)\\to \\infty$ as $n\\to \\infty$ such that, for all large $n$, there is a composite number $m$ such that $ n+f(n)n}(m-p(m)), $ and whether $n-F(n)\\sim cn^{1/2}$ for some $c>0$.\nSee also [385].\nReferences\n\n\n[Er92e] Erdos, P\\'{a}l, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2132, "problem_number": "EP-467", "title": "Erdős Problem #467", "statement": "Prove the following for all large $x$: there is a choice of congruence classes $a_p$ for all primes $p\\leq x$ and a decomposition $\\{p\\leq x\\}=A\\sqcup B$ into two non-empty sets such that, for all $n1$, so the restriction $k\neq 1$ is necessary. Erdos and Graham report that Graham, Lehmer, and Lehmer have proved this for $k=2^i$ for $i\\geq 1$, or if $k=-1$, but I cannot find such a paper. Tang has written a short note giving a proof for this case.\nAs an indication of the difficulty, when $k=3$ the smallest $n$ such that $2^n\\equiv 3\\pmod{n}$ is $n=4700063497$.\nThe minimal such $n$ for each $k$ is A036236 in the OEIS.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2140, "problem_number": "EP-483", "title": "Erdős Problem #483", "statement": "Let $f(k)$ be the minimal $N$ such that if $\\{1,\\ldots,N\\}$ is $k$-coloured then there is a monochromatic solution to $a+b=c$. Estimate $f(k)$. In particular, is it true that $f(k) < c^k$ for some constant $c>0$?", "background": "The values of $f(k)$ are known as Schur numbers. The best-known bounds for large $k$ are $ (380)^{k/5}-O(1)\\leq f(k) \\leq \\lfloor(e-\\tfrac{1}{24}) k!\\rfloor-1. $ The lower bound is due to Ageron, Casteras, Pellerin, Portella, Rimmel, and Tomasik \\cite{ACPPRT21} (improving previous bounds of Exoo \\cite{Ex94} and Fredricksen and Sweet \\cite{FrSw00}) and the upper bound is due to Whitehead \\cite{Wh73}. Note that $380^{1/5}\\approx 3.2806$.\nThe known values of $f$ are $f(1)=2$, $f(2)=5$, $f(3)=14$, $f(4)=45$, and $f(5)=161$ (see A030126). (The equality $f(5)=161$ was established by Heule \\cite{He17}).\nSee also [183] (in particular a folklore observation gives $f(k)\\leq R(3;k)-1$).\nReferences\n\n\n[ACPPRT21] R. Ageron, P. Casteras, T. Pellerin, Y. Portella, A. Rimmel, and J. Tomasik, New lower bounds for Schur and weak Schur numbers. arXiv:2112.03175 (2021).\n\n[Ex94] Exoo, G., A lower bound for Schur numbers and multicolor Ramsey numbers. Electronic J. of Combinatorics (1994).\n\n[FrSw00] Fredricksen, Harold and Sweet, Melvin M., Symmetric sum-free partitions and lower bounds for {S}chur\nnumbers. Electron. J. Combin. (2000), Research Paper 32, 9.\n\n[He17] M. Heuele, Schur Number Five. arXiv:1711.08076 (2017).\n\n[Wh73] Whitehead, Jr., Earl Glen, The {R}amsey number {$N(3,\\,3,\\,3,\\,3;\\,2)$}. Discrete Math. (1973), 389--396.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2141, "problem_number": "EP-486", "title": "Erdős Problem #486", "statement": "Let $A\\subseteq \\mathbb{N}$, and for each $n\\in A$ choose some $X_n\\subseteq \\mathbb{Z}/n\\mathbb{Z}$. Let $ B = \\{ m\\in \\mathbb{N} : m\not\\in X_n\\pmod{n}\\textrm{ for all }n\\in A\\textrm{ with }m>n\\}. $ Must $B$ have a logarithmic density, i.e. is it true that $ \\lim_{x\\to \\infty} \\frac{1}{\\log x}\\sum_{\\substack{m\\in B\\\\ mn\\geq \\max(A)$, $ \\frac{\\lvert B\\cap [1,m]\\rvert }{m}< 2\\frac{\\lvert B\\cap [1,n]\\rvert}{n}? $ ", "background": "The constant $2$ would be the best possible here, as witnessed by taking $A=\\{a\\}$, $n=2a-1$, and $m=2a$.\nThis problem is also discussed in problem E5 of Guy's collection \\cite{Gu04}.\nIn \\cite{Er61} this problem is as stated above, but with $a\\mid n$ in the definition of $B$ replaced by $a\nmid n$. This is most likely a typo (especially since the problem is also given as stated above in \\cite{Er66}). There have been several counterexamples given for this alternate problem. Cambie has observed that, if $A$ is the set of primes bounded above by $n$, and $m=2n$, then $ \\frac{\\lvert B\\cap [1,m]\\rvert }{m}=\\frac{\\pi(2n)-\\pi(n)+1}{2n}\\sim \\frac{1}{2\\log n} $ while $ \\frac{\\lvert B\\cap [1,n]\\rvert}{n}=\\frac{1}{n}. $ Further concrete counterexamples, found by Alexeev and Aristotle, are given in the comments section.\nReferences\n\n\n[Er61] Erdos, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutat\\'{o} Int. K\"{o}zl. (1961), 221-254.\n\n[Er66] Erdos, P\\'al, Remarks on number theory. {V}. {E}xtremal problems in number\ntheory. {II}. Mat. Lapok (1966), 135--155.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2143, "problem_number": "EP-489", "title": "Erdős Problem #489", "statement": "Let $A\\subseteq \\mathbb{N}$ be a set such that $\\lvert A\\cap [1,x]\\rvert=o(x^{1/2})$. Let $ B=\\{ n\\geq 1 : a\nmid n\\textrm{ for all }a\\in A\\}. $ If $B=\\{b_10$ and $\\theta$ such that $ \\sum_{n\\in A}\\cos(n\\theta) < -cN^{1/2}? $ ", "background": "Chowla's cosine problem. Ruzsa \\cite{Ru04} (improving on an earlier result of Bourgain \\cite{Bo86}), proved an upper bound of $ -\\exp(O(\\sqrt{\\log N})). $ Polynomial bounds were proved independently by Bedert \\cite{Be25c} and Jin, Milojevi\\'{c}, Tomon, and Zhang \\cite{JMTZ25}. The best bound follows from the method of Bedert \\cite{Be25c}, which proved the existence of some $c>0$ such that, for all $A$ of size $N$, $ \\sum_{n\\in A}\\cos(n\\theta) < -cN^{1/7}. $ The example $A=B-B$, where $B$ is a Sidon set, shows that $N^{1/2}$ would be the best possible here.\nThis problem is Problem 81 on Green's open problems list.\nThis is related to [256].\nReferences\n\n\n[Be25c] B. Bedert, Polynomial bounds for the Chowla Cosine Problem. arXiv:2509.05260 (2025).\n\n[Bo86] Bourgain, J., Sur le minimum d'une somme de cosinus. Acta Arith. (1986), 381-389.\n\n[JMTZ25] Z. Jin, A. Milojevi\\'{c}, I. Tomon, and S. Zhang, From small eigenvalues to large cuts, and Chowla's cosine problem. arXiv:2509.03490 (2025).\n\n[Ru04] Ruzsa, Imre Z., Negative values of cosine sums. Acta Arith. (2004), 179-186.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2152, "problem_number": "EP-513", "title": "Erdős Problem #513", "statement": "Let $f=\\sum_{n=0}^\\infty a_nz^n$ be a transcendental entire function. What is the greatest possible value of $ \\liminf_{r\\to \\infty} \\frac{\\max_n\\lvert a_nr^n\\rvert}{\\max_{\\lvert z\\rvert=r}\\lvert f(z)\\rvert}? $ ", "background": "It is trivial that this value is in $[1/2,1)$. K\"{o}v\\'{a}ri (unpublished) observed that it must be $>1/2$. Clunie and Hayman \\cite{ClHa64} showed that it is $\\leq 2/\\pi-c$ for some absolute constant $c>0$. Some other results on this quantity were established by Gray and Shah \\cite{GrSh63}.\nSee also [227].\nReferences\n\n\n[ClHa64] Clunie, J. and Hayman, W. K., The maximum term of a power series. J. Analyse Math. (1964), 143-186.\n\n[GrSh63] Gray, Alfred and Shah, S. M., A note on entire functions and a conjecture of Erdos. Bull. Amer. Math. Soc. (1963), 573-577.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2153, "problem_number": "EP-514", "title": "Erdős Problem #514", "statement": "Let $f(z)$ be an entire transcendental function. Does there exist a path $L$ so that, for every $n$, $ \\lvert f(z)/z^n\\rvert \\to \\infty $ as $z\\to \\infty$ along $L$?\nCan the length of this path be estimated in terms of $M(r)=\\max_{\\lvert z\\rvert=r}\\lvert f(z)\\rvert$? Does there exist a path along which $\\lvert f(z)\\rvert$ tends to $\\infty$ faster than a fixed function of $M(r)$ (such that $M(r)^\\epsilon$)?", "background": "Boas (unpublished) has proved the first part, that such a path must exist.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2154, "problem_number": "EP-517", "title": "Erdős Problem #517", "statement": "Let $f(z)=\\sum_{k=1}^\\infty a_kz^{n_k}$ be an entire function (with $a_k\neq 0$ for all $k\\geq 1$). Is it true that if $n_k/k\\to \\infty$ then $f(z)$ assumes every value infinitely often?", "background": "A conjecture of Fej\\'{e}r and P\\'{o}lya.\nFej\\'{e}r \\cite{Fe08} proved that if $\\sum\\frac{1}{n_k}<\\infty$ then $f(z)$ assumes every value at least once, and Biernacki \\cite{Bi28} proved that if $\\sum\\frac{1}{n_k}<\\infty$ then $f(z)$ assumes every value infinitely often.\nP\\'{o}lya \\cite{Po29} proved that if $f$ has finite order then $f(z)$ assumes every value infinitely often under the assumption that $\\limsup (n_{k+1}-n_k)=\\infty$.\nReferences\n\n\n[Bi28] Biernacki, Mi\\'{e}cislas, Sur les \\'{e}quations alg\\'{e}briques contenant des param\\'{e}tres arbitraires. (1928), 145.\n\n[Fe08] Fej\\'{e}r, Leopold, \"{U}ber die Wurzel vom kleinsten absoluten Betrage einer algebraischen Gleichung. Math. Ann. (1908), 413-423.\n\n[Po29] P\\'olya, G., Untersuchungen \"uber {L}\"ucken und {S}ingularit\"{a}ten von\n{P}otenzreihen. Math. Z. (1929), 549--640.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2155, "problem_number": "EP-520", "title": "Erdős Problem #520", "statement": "Let $f$ be a Rademacher multiplicative function: a random $\\{-1,0,1\\}$-valued multiplicative function, where for each prime $p$ we independently choose $f(p)\\in \\{-1,1\\}$ uniformly at random, and for square-free integers $n$ we extend $f(p_1\\cdots p_r)=f(p_1)\\cdots f(p_r)$ (and $f(n)=0$ if $n$ is not squarefree). Does there exist some constant $c>0$ such that, almost surely, $ \\limsup_{N\\to \\infty}\\frac{\\sum_{m\\leq N}f(m)}{\\sqrt{N\\log\\log N}}=c? $ ", "background": "Note that if we drop the multiplicative assumption, and simply assign $f(m)=\\pm 1$ at random, then this statement is true (with $c=\\sqrt{2}$), the law of the iterated logarithm.\nWintner \\cite{Wi44} proved that, almost surely, $ \\sum_{m\\leq N}f(m)\\ll N^{1/2+o(1)}, $ and Erdos improved the right-hand side to $N^{1/2}(\\log N)^{O(1)}$. Lau, Tenenbaum, and Wu \\cite{LTW13} have shown that, almost surely, $ \\sum_{m\\leq N}f(m)\\ll N^{1/2}(\\log\\log N)^{2+o(1)}. $ Caich \\cite{Ca24b} has improved this to $ \\sum_{m\\leq N}f(m)\\ll N^{1/2}(\\log\\log N)^{3/4+o(1)}. $ Harper \\cite{Ha13} has shown that the sum is almost surely not $O(N^{1/2}/(\\log\\log N)^{5/2+o(1)})$, and conjectured that in fact Erdos' conjecture is false, and almost surely $ \\sum_{m\\leq N}f(m) \\ll N^{1/2}(\\log\\log N)^{1/4+o(1)}. $ \nReferences\n\n\n[Ca24b] R. Caich, Almost sure upper bound for random multiplicative functions. arXiv:2304.00943 (2024).\n\n[Ha13] Harper, Adam J., Bounds on the suprema of Gaussian processes, and omega\nresults for the sum of a random multiplicative function. Ann. Appl. Probab. (2013), 584-616.\n\n[LTW13] Lau, Yuk-Kam and Tenenbaum, G\\'{e}rald and Wu, Jie, On mean values of random multiplicative functions. Proc. Amer. Math. Soc. (2013), 409-420.\n\n[Wi44] Wintner, Aurel, Random factorizations and Riemann's hypothesis. Duke Math. J. (1944), 267-275.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2156, "problem_number": "EP-521", "title": "Erdős Problem #521", "statement": "Let $(\\epsilon_k)_{k\\geq 0}$ be independently uniformly chosen at random from $\\{-1,1\\}$. If $R_n$ counts the number of real roots of $f_n(z)=\\sum_{0\\leq k\\leq n}\\epsilon_k z^k$ then is it true that, almost surely, $ \\lim_{n\\to \\infty}\\frac{R_n}{\\log n}=\\frac{2}{\\pi}? $ ", "background": "Erdos and Offord \\cite{EO56} showed that the number of real roots of a random degree $n$ polynomial with $\\pm 1$ coefficients is $(\\frac{2}{\\pi}+o(1))\\log n$.\nIt is ambiguous in \\cite{Er61} whether Erdos intended the coefficients to be uniformly chosen from $\\{-1,1\\}$ or $\\{0,1\\}$. In the latter case, the constant $\\frac{2}{\\pi}$ should be $\\frac{1}{\\pi}$ (see the discussion in the comments).\nIn the case of $\\{-1,1\\}$ Do \\cite{Do24} proved that, if $R_n[-1,1]$ counts the number of roots in $[-1,1]$, then, almost surely, $ \\lim_{n\\to \\infty}\\frac{R_n[-1,1]}{\\log n}=\\frac{1}{\\pi}. $ See also [522].\nReferences\n\n\n[Do24] Y. Do, A strong law of large numbers for real roots of random polynomials. arXiv:2403.06353 (2024).\n\n[EO56] Erd\"{o}s, Paul and Offord, A. C., On the number of real roots of a random algebraic equation. Proc. London Math. Soc. (3) (1956), 139-160.\n\n[Er61] Erdos, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutat\\'{o} Int. K\"{o}zl. (1961), 221-254.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2157, "problem_number": "EP-522", "title": "Erdős Problem #522", "statement": "Let $f(z)=\\sum_{0\\leq k\\leq n} \\epsilon_k z^k$ be a random polynomial, where $\\epsilon_k\\in \\{-1,1\\}$ independently uniformly at random for $0\\leq k\\leq n$.\nIs it true that, if $R_n$ is the number of roots of $f(z)$ in $\\{ z\\in \\mathbb{C} : \\lvert z\\rvert \\leq 1\\}$, then $ \\frac{R_n}{n/2}\\to 1 $ almost surely?", "background": "Random polynomials with independently identically distributed coefficients are sometimes called Kac polynomials - this problem considers the case of Rademacher coefficients, i.e. independent uniform $\\pm 1$ values. Erdos and Offord \\cite{EO56} showed that the number of real roots of a random degree $n$ polynomial with $\\pm 1$ coefficients is $(\\frac{2}{\\pi}+o(1))\\log n$.\nThere is some ambiguity whether Erdos intended the coefficients to be in $\\{-1,1\\}$ or $\\{0,1\\}$ - see the comments section.\nA weaker version of this was solved by Yakir \\cite{Ya21}, who proved that $ \\frac{R_n}{n/2}\\to 1 $ in probability. (This weaker claim was also asked by Erdos, and also appears in a book of Hayman \\cite{Ha67}.) More precisely, $ \\lim_{n\\to \\infty} \\mathbb{P}(\\lvert R_n-n/2\\rvert \\geq n^{9/10}) =0. $ See also [521].\nReferences\n\n\n[EO56] Erd\"{o}s, Paul and Offord, A. C., On the number of real roots of a random algebraic equation. Proc. London Math. Soc. (3) (1956), 139-160.\n\n[Ha67] Hayman, W. K., Research problems in function theory. (1967), vii+56.\n\n[Ya21] Yakir, Oren, Approximately half of the roots of a random {L}ittlewood\npolynomial are inside the disk. Studia Math. (2021), 227--240.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2158, "problem_number": "EP-524", "title": "Erdős Problem #524", "statement": "For any $t\\in (0,1)$ let $t=\\sum_{k=1}^\\infty \\epsilon_k(t)2^{-k}$ (where $\\epsilon_k(t)\\in \\{0,1\\}$). What is the correct order of magnitude (for almost all $t\\in(0,1)$) for $ M_n(t)=\\max_{x\\in [-1,1]}\\left\\lvert \\sum_{k\\leq n}(-1)^{\\epsilon_k(t)}x^k\\right\\rvert? $ ", "background": "A problem of Salem and Zygmund \\cite{SaZy54}. Chung showed that, for almost all $t$, there exist infinitely many $n$ such that $ M_n(t) \\ll \\left(\\frac{n}{\\log\\log n}\\right)^{1/2}. $ Erdos (unpublished) showed that for almost all $t$ and every $\\epsilon>0$ we have $\\lim_{n\\to \\infty}M_n(t)/n^{1/2-\\epsilon}=\\infty$.\nReferences\n\n\n[SaZy54] Salem, R. and Zygmund, A., Some properties of trigonometric series whose terms have\nrandom signs. Acta Math. (1954), 245-301.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2159, "problem_number": "EP-528", "title": "Erdős Problem #528", "statement": "Let $f(n,k)$ count the number of self-avoiding walks of $n$ steps (beginning at the origin) in $\\mathbb{Z}^k$ (i.e. those walks which do not intersect themselves). Determine $ C_k=\\lim_{n\\to\\infty}f(n,k)^{1/n}. $ ", "background": "The constant $C_k$ is sometimes known as the connective constant. Hammersley and Morton \\cite{HM54} showed that this limit exists, and it is trivial that $k\\leq C_k\\leq 2k-1$.\nKesten \\cite{Ke63} proved that $C_k=2k-1-1/2k+O(1/k^2)$, and more precise asymptotics are given by Clisby, Liang, and Slade \\cite{CLS07}.\nConway and Guttmann \\cite{CG93} showed that $C_2\\geq 2.62$ and Alm \\cite{Al93} showed that $C_2\\leq 2.696$. Jacobsen, Scullard, and Guttmann \\cite{JSG16} have computed the first few decimal places of $C_2$, showing that $ C_2 = 2.6381585303279\\cdots. $ See also [529].\nReferences\n\n\n[Al93] Alm, Sven Erick, Upper bounds for the connective constant of self-avoiding\nwalks. Combin. Probab. Comput. (1993), 115-136.\n\n[CG93] Conway, A. R. and Guttmann, A. J., Lower bound on the connective constant for square lattice\nself-avoiding walks. J. Phys. A (1993), 3719-3724.\n\n[CLS07] Clisby, Nathan and Liang, Richard and Slade, Gordon, Self-avoiding walk enumeration via the lace expansion. J. Phys. A (2007), 10973-11017.\n\n[HM54] Hammersley, J. M. and Morton, K. W., Poor man's Monte Carlo. J. Roy. Statist. Soc. Ser. B (1954), 23-38; discussion 61-75.\n\n[JSG16] Jacobsen, Jesper Lykke and Scullard, Christian R. and\nGuttmann, Anthony J., On the growth constant for square-lattice self-avoiding walks. J. Phys. A (2016), 494004, 18.\n\n[Ke63] Kesten, Harry, On the number of self-avoiding walks. J. Mathematical Phys. (1963), 960-969.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2160, "problem_number": "EP-529", "title": "Erdős Problem #529", "statement": "Let $d_k(n)$ be the expected distance from the origin after taking $n$ random steps from the origin in $\\mathbb{Z}^k$ (conditional on no self intersections) - that is, a self-avoiding walk. Is it true that $ \\lim_{n\\to \\infty}\\frac{d_2(n)}{n^{1/2}}= \\infty? $ Is it true that $ d_k(n)\\ll n^{1/2} $ for $k\\geq 3$?", "background": "Slade \\cite{Sl87} proved that, for $k$ sufficiently large, $d_k(n)\\sim Dn^{1/2}$ for some constant $D>0$ (independent of $k$). Hara and Slade (\\cite{HaSl91} and \\cite{HaSl92}) proved this for all $k\\geq 5$.\nFor $k=2$ Duminil-Copin and Hammond \\cite{DuHa13} have proved that $d_2(n)=o(n)$.\nIt is now conjectured that $d_k(n)\\ll n^{1/2}$ is false for $k=3$ and $k=4$, and more precisely (see for example Section 1.4 of \\cite{MaSl93}) that $d_2(n)\\sim Dn^{3/4}$, $d_3(n)\\sim n^{\nu}$ where $\nu\\approx 0.59$, and $d_4(n)\\sim D(\\log n)^{1/8}n^{1/2}$.\nMadras and Slade \\cite{MaSl93} have a monograph on the topic of self-avoiding walks.\nSee also [528].\nReferences\n\n\n[DuHa13] Duminil-Copin, Hugo and Hammond, Alan, Self-avoiding walk is sub-ballistic. Comm. Math. Phys. (2013), 401--423.\n\n[HaSl91] Hara, Takashi and Slade, Gordon, Critical behaviour of self-avoiding walk in five or more\ndimensions. Bull. Amer. Math. Soc. (N.S.) (1991), 417--423.\n\n[HaSl92] Hara, Takashi and Slade, Gordon, Self-avoiding walk in five or more dimensions. {I}. {T}he\ncritical behaviour. Comm. Math. Phys. (1992), 101--136.\n\n[MaSl93] Madras, Neal and Slade, Gordon, The self-avoiding walk. (1993), xiv+425.\n\n[Sl87] Slade, Gordon, The diffusion of self-avoiding random walk in high dimensions. Comm. Math. Phys. (1987), 661--683.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2161, "problem_number": "EP-530", "title": "Erdős Problem #530", "statement": "Let $\\ell(N)$ be maximal such that in any finite set $A\\subset \\mathbb{R}$ of size $N$ there exists a Sidon subset $S$ of size $\\ell(N)$ (i.e. the only solutions to $a+b=c+d$ in $S$ are the trivial ones). Determine the order of $\\ell(N)$.", "background": "In particular, is it true that $\\ell(N)\\sim N^{1/2}$?\nOriginally asked by Riddell \\cite{Ri69}. Erdos noted the bounds $ N^{1/3} \\ll \\ell(N) \\leq (1+o(1))N^{1/2} $ (the upper bound following from the case $A=\\{1,\\ldots,N\\}$). The lower bound was improved to $N^{1/2}\\ll \\ell(N)$ by Koml\\'{o}s, Sulyok, and Szemer\\'{e}di \\cite{KSS75}. The correct constant is unknown, but it is likely that the upper bound is true, so that $\\ell(N)\\sim N^{1/2}$.\nIn \\cite{AlEr85} Alon and Erdos make the stronger conjecture that perhaps $A$ can always be written as the union of at most $(1+o(1))N^{1/2}$ many Sidon sets. (This is easily verified for $A=\\{1,\\ldots,N\\}$ using standard constructions of Sidon sets.)\nThis is discussed in problem C9 of Guy's collection \\cite{Gu04}.\nSee also [1088] for a higher-dimensional generalisation.\nReferences\n\n\n[AlEr85] Alon, Noga and Erdos, P., An application of graph theory to additive number theory. European J. Combin. (1985), 201-203.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[KSS75] Koml\\'{o}s, J. and Sulyok, M. and Szemeredi, E., Linear problems in combinatorial number theory. Acta Math. Acad. Sci. Hungar. (1975), 113-121.\n\n[Ri69] Riddell, J., On sets of numbers containing no $l$ terms in arithmetic progression. Nieuw Arch. Wisk. (3) (1969), 204-209.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2162, "problem_number": "EP-531", "title": "Erdős Problem #531", "statement": "Let $F(k)$ be the minimal $N$ such that if we two-colour $\\{1,\\ldots,N\\}$ there is a set $A$ of size $k$ such that all subset sums $\\sum_{a\\in S}a$ (for $\\emptyset\neq S\\subseteq A$) are monochromatic. Estimate $F(k)$.", "background": "The existence of $F(k)$ was established by Sanders and Folkman, and it also follows from Rado's theorem. It is commonly known as Folkman's theorem.\nErdos and Spencer \\cite{ErSp89} proved that $ F(k) \\geq 2^{ck^2/\\log k} $ for some constant $c>0$. Balogh, Eberhrad, Narayanan, Treglown, and Wagner \\cite{BENTW17} have improved this to $ F(k) \\geq 2^{2^{k-1}/k}. $ \nReferences\n\n\n[BENTW17] Balogh, J\\'{o}zsef and Eberhard, Sean and Narayanan, Bhargav and Treglown, Andrew and Wagner, Adam Zsolt, An improved lower bound for Folkman's theorem. Bull. Lond. Math. Soc. (2017), 745-747.\n\n[ErSp89] Erdos, Paul and Spencer, Joel, Monochromatic sumsets. J. Combin. Theory Ser. A (1989), 162-163.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2163, "problem_number": "EP-533", "title": "Erdős Problem #533", "statement": "Let $\\delta>0$. If $n$ is sufficiently large and $G$ is a graph on $n$ vertices with no $K_5$ and at least $\\delta n^2$ edges then $G$ contains a set of $\\gg_\\delta n$ vertices containing no triangle.", "background": "A problem of Erdos, Hajnal, Simonovits, S\\'{o}s, and Szemer\\'{e}di, who could prove this is true for $\\delta>1/16$, and could further prove it for $\\delta>0$ if we replace $K_5$ with $K_4$.\nThey further observed that it fails for $\\delta =1/4$ if we replace $K_5$ with $K_7$: by a construction of Erdos and Rogers \\cite{ErRo62} (see [620]) there exists some constant $c>0$ such that, for all large $n$, there is a graph on $n$ vertices which contains no $K_4$ and every set of at least $n^{1-c}$ vertices contains a triangle. If we take two vertex disjoint copies of this graph and add all edges between the two copies then this yields a graph on $2n$ vertices with $\\geq n^2$ edges, which contains no $K_7$, yet every set of at least $2n^{1-c}$ vertices contains a triangle.\nSee also [579] and the entry in the graphs problem collection.\nReferences\n\n\n[ErRo62] Erdos, P. and Rogers, C. A., The construction of certain graphs. Canadian J. Math. (1962), 702-707.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2164, "problem_number": "EP-535", "title": "Erdős Problem #535", "statement": "Let $r\\geq 3$, and let $f_r(N)$ denote the size of the largest subset of $\\{1,\\ldots,N\\}$ such that no subset of size $r$ has the same pairwise greatest common divisor between all elements. Estimate $f_r(N)$.", "background": "Erdos \\cite{Er64} proved that $ f_r(N) \\leq N^{\\frac{3}{4}+o(1)}, $ and Abbott and Hanson \\cite{AbHa70} improved this exponent to $1/2$. Erdos \\cite{Er64} proved the lower bound $ f_3(N) > N^{\\frac{c}{\\log\\log N}} $ for some constant $c>0$, and conjectured this should also be an upper bound.\nErdos writes this is 'intimately connected' with the sunflower problem [20]. Indeed, the conjectured upper bound would follow from the following stronger version of the sunflower problem: estimate the size of the largest set of integers $A$ such that $\\omega(n)=k$ for all $n\\in A$ and there does not exist $a_1,\\ldots,a_r\\in A$ and an integer $d$ such that $(a_i,a_j)=d$ for all $i\neq j$ and $(a_i/d,d)=1$ for all $i$. The conjectured upper bound for $f_r(N)$ would follow if the size of such an $A$ must be at most $c_r^k$. The original sunflower proof of Erdos and Rado gives the upper bound $c_r^kk!$.\nSee also [536].\nReferences\n\n\n[AbHa70] Abbott, H. L. and Hanson, D., An extremal problem in number theory. Bull. London Math. Soc. (1970), 324-326.\n\n[Er64] Erdos, P., On a problem in elementary number theory and a combinatorial problem. Math. Comp. (1964), 644-646.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2165, "problem_number": "EP-536", "title": "Erdős Problem #536", "statement": "Let $\\epsilon>0$ and $N$ be sufficiently large. Is it true that if $A\\subseteq \\{1,\\ldots,N\\}$ has size at least $\\epsilon N$ then there must be distinct $a,b,c\\in A$ such that $ [a,b]=[b,c]=[a,c], $ where $[a,b]$ denotes the least common multiple?", "background": "This is false if we ask for four elements with the same pairwise least common multiple, as shown by Erdos \\cite{Er62} (with a proof given in \\cite{Er70}).\nThis was also asked by Erdos at the 1991 problem session of West Coast Number Theory.\nIn the comments Weisenberg sketches a construction of a set $A\\subseteq [1,N]$ without this property such that $ \\lvert A\\rvert \\gg (\\log\\log N)^{f(N)}\\frac{N}{\\log N} $ for some $f(N)\\to \\infty$. Weisenberg also sketches a proof of the main problem when $\\epsilon>\\frac{221}{225}$.\nSee also [535], [537], and [856]. A related combinatorial problem is asked at [857].\nReferences\n\n\n[Er62] Erdos, P\\'{a}l, Remarks on number theory. IV. Extremal problems in number theory. I. Mat. Lapok (1962), 228-255.\n\n[Er70] Erdos, Paul, Some extremal problems in combinatorial number theory. Mathematical Essays Dedicated to A. J. Macintyre (1970), 123-133.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2166, "problem_number": "EP-538", "title": "Erdős Problem #538", "statement": "Let $r\\geq 2$ and suppose that $A\\subseteq\\{1,\\ldots,N\\}$ is such that, for any $m$, there are at most $r$ solutions to $m=pa$ where $p$ is prime and $a\\in A$. Give the best possible upper bound for $ \\sum_{n\\in A}\\frac{1}{n}. $ ", "background": "Erdos observed that $ \\sum_{n\\in A}\\frac{1}{n}\\sum_{p\\leq N}\\frac{1}{p}\\leq r\\sum_{m\\leq N^2}\\frac{1}{m}\\ll r\\log N, $ and hence $ \\sum_{n\\in A}\\frac{1}{n} \\ll r\\frac{\\log N}{\\log\\log N}. $ See also [536] and [537].\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2167, "problem_number": "EP-539", "title": "Erdős Problem #539", "statement": "Let $h(n)$ be such that, for any set $A\\subseteq \\mathbb{N}$ of size $n$, the set $ \\left\\{ \\frac{a}{(a,b)}: a,b\\in A\\right\\} $ has size at least $h(n)$. Estimate $h(n)$.", "background": "Erdos and Szemer\\'{e}di proved that $ n^{1/2} \\ll h(n) \\ll n^{1-c} $ for some constant $c>0$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2168, "problem_number": "EP-543", "title": "Erdős Problem #543", "statement": "Define $f(N)$ be the minimal $k$ such that the following holds: if $G$ is an abelian group of size $N$ and $A\\subseteq G$ is a random set of size $k$ then, with probability $\\geq 1/2$, all elements of $G$ can be written as $\\sum_{x\\in S}x$ for some $S\\subseteq A$. Is $ f(N) \\leq \\log_2 N+o(\\log\\log N)? $ ", "background": "Erdos and R\\'{e}nyi \\cite{ErRe65} proved that $ f(N) \\leq \\log_2N+O(\\log\\log N). $ Erdos believed improving this to $o(\\log\\log N)$ is impossible.\nReferences\n\n\n[ErRe65] Erdos, P. and R\\'{e}nyi, A., Probabilistic methods in group theory. J. Analyse Math. (1965), 127-138.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2169, "problem_number": "EP-544", "title": "Erdős Problem #544", "statement": "Show that $ R(3,k+1)-R(3,k)\\to\\infty $ as $k\\to \\infty$. Similarly, prove or disprove that $ R(3,k+1)-R(3,k)=o(k). $ ", "background": "A problem of Erdos and S\\'{o}s.\nThis problem is #8 in Ramsey Theory in the graphs problem collection.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2170, "problem_number": "EP-545", "title": "Erdős Problem #545", "statement": "Let $G$ be a graph with $m$ edges and no isolated vertices. Is the Ramsey number $R(G)$ maximised when $G$ is 'as complete as possible'? That is, if $m=\\binom{n}{2}+t$ edges with $0\\leq t0$, there are infinitely many $n$ such that $ R(C_4,S_n)\\leq n+\\sqrt{n}-c? $ ", "background": "A problem of Burr, Erdos, Faudree, Rousseau, and Schelp \\cite{BEFRS89}. Erdos often asked about $R(C_4,S_n)$ in the equivalent formulation of asking for a bound on the minimum degree of a graph which would guarantee the existence of a $C_4$ (see [85]).\nIt is known that $ n+\\sqrt{n}-6n^{11/40} \\leq R(C_4,S_n)\\leq n+\\lceil\\sqrt{n}\\rceil+1. $ The lower bound is due to \\cite{BEFRS89}, the upper bound is due to Parsons \\cite{Pa75}. The lower bound of \\cite{BEFRS89} is related to gaps between primes, and assuming e.g. Cramer's conjecture on gaps between primes their lower bound would be $n+\\sqrt{n}-n^{o(1)}$.\nErdos offered \\$100 for a proof or disproof of the second question in \\cite{BEFRS89}. In \\cite{Er96} Erdos asks (an equivalent formulation of) whether $R(C_4,S_n)\\geq n+\\sqrt{n}-O(1)$, but says this is probably 'too optimistic'.\nThey also ask, if $f(n)=R(C_4,S_n)$, whether $f(n+1)=f(n)$ infinitely often, and is the density of such $n$ $0$? Also, is it true that $f(n+1)\\leq f(n)+2$ for all $n$? A similar question about an equivalent function is the subject of [85].\nParsons \\cite{Pa75} proved that $ R(C_4,S_n)=n+\\lceil\\sqrt{n}\\rceil $ whenever $n=q^2+1$ for a prime power $q$ and $ R(C_4,S_n)=n+\\lceil\\sqrt{n}\\rceil+1 $ whenever $n=q^2$ for a prime power $q$ (in particular both equalities occur infinitely often).\nThis has been extended in various works, all in the cases $n=q^2\\pm t$ for some $0\\leq t\\leq q$ and prime power $q$. We refer to the work of Parsons \\cite{Pa76}, Wu, Sun, Zhang, and Radziszowski \\cite{WSZR15}, and Zhang, Chen, and Cheng (\\cite{ZCC17} and \\cite{ZCC17b}) for a precise description. In every known case $ R(C_4,S_n)=n+\\lceil\\sqrt{n}\\rceil+\\{0,1\\}, $ and Zhang, Chen, and Cheng \\cite{ZCC17} speculate whether this is in fact true for all $n\\geq 2$ (whence the answer to the question above would be no).\nThis problem is #19 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[BEFRS89] Burr, S. and Erd\"{o}s, P. and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., Some complete bipartite graph-tree Ramsey numbers. Graph theory in memory of G. A. Dirac (Sandbjerg,\n1985) (1989), 79-89.\n\n[Er96] Erdos, Paul, Some of my favourite problems on cycles and colourings. Tatra Mt. Math. Publ. (1996), 7-9.\n\n[Pa75] Parsons, T. D., Ramsey graphs and block designs. {I}. Trans. Amer. Math. Soc. (1975), 33--44.\n\n[Pa76] No reference found.\n\n\n[WSZR15] Wu, Yali and Sun, Yongqi and Zhang, Rui and Radziszowski,\nStanis\\l aw P., Ramsey numbers of {$C_4$} versus wheels and stars. Graphs Combin. (2015), 2437--2446.\n\n[ZCC17] Zhang, Xuemei and Chen, Yaojun and Cheng, T. C. Edwin, Some values of {R}amsey numbers for {$C_4$} versus stars. Finite Fields Appl. (2017), 73--85.\n\n[ZCC17b] Zhang, Xuemei and Chen, Yaojun and Cheng, T. C. Edwin, Polarity graphs and {R}amsey numbers for {$C_4$} versus stars. Discrete Math. (2017), 655--660.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2173, "problem_number": "EP-554", "title": "Erdős Problem #554", "statement": "Let $R(G;k)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Show that $ \\lim_{k\\to \\infty}\\frac{R(C_{2n+1};k)}{R(K_3;k)}=0 $ for any $n\\geq 2$.", "background": "A problem of Erdos and Graham. The problem is open even for $n=2$.\nThis problem is #23 in Ramsey Theory in the graphs problem collection.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2174, "problem_number": "EP-555", "title": "Erdős Problem #555", "statement": "Let $R(G;k)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Determine the value of $ R(C_{2n};k). $ ", "background": "A problem of Erdos and Graham. Erdos \\cite{Er81c} gives the bounds $ k^{1+\\frac{1}{2n}}\\ll R(C_{2n};k)\\ll k^{1+\\frac{1}{n-1}}. $ Chung and Graham \\cite{ChGr75} showed that $ R(C_4;k)>k^2-k+1 $ when $k-1$ is a prime power and $ R(C_4;k)\\leq k^2+k+1 $ for all $k$.\nThis problem is #24 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[ChGr75] Chung, Fan R. K. and Graham, R. L., On multicolor Ramsey numbers for complete bipartite graphs. J. Combinatorial Theory Ser. B (1975), 164-169.\n\n[Er81c] Erdos, Paul, Some new problems and results in graph theory and other branches of combinatorial mathematics. Combinatorics and graph theory (1981), 9-17.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2175, "problem_number": "EP-557", "title": "Erdős Problem #557", "statement": "Let $R(G;k)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Is it true that $ R(T;k)\\leq kn+O(1) $ for any tree $T$ on $n$ vertices?", "background": "A problem of Erdos and Graham. Implied by [548].\nThis would be best possible since, for example, $R(S_n,k)\\geq kn-O(k)$ if $S_n=K_{1,n-1}$ is a star on $n$ vertices.\nThis problem is #26 in Ramsey Theory in the graphs problem collection.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2176, "problem_number": "EP-558", "title": "Erdős Problem #558", "statement": "Let $R(G;k)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Determine $ R(K_{s,t};k) $ where $K_{s,t}$ is the complete bipartite graph with $s$ vertices in one component and $t$ in the other.", "background": "Chung and Graham \\cite{ChGr75} prove the general bounds $ (2\\pi\\sqrt{st})^{\\frac{1}{s+t}}\\left(\\frac{s+t}{e^2}\\right)k^{\\frac{st-1}{s+t}}\\leq R(K_{s,t};k)\\leq (t-1)(k+k^{1/s})^s $ and determined $ R(K_{2,2},k)=(1+o(1))k^2. $ Alon, R\\'{o}nyai, and Szab\\'{o} \\cite{ARS99} have proved that $ R(K_{3,3},k)=(1+o(1))k^3 $ and that if $s\\geq (t-1)!+1$ then $ R(K_{s,t},k)\\asymp k^t. $ This problem is #27 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[ARS99] Alon, Noga and R\\'{o}nyai, Lajos and Szab\\'{o}, Tibor, Norm-graphs: variations and applications. J. Combin. Theory Ser. B (1999), 280-290.\n\n[ChGr75] Chung, Fan R. K. and Graham, R. L., On multicolor Ramsey numbers for complete bipartite graphs. J. Combinatorial Theory Ser. B (1975), 164-169.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2177, "problem_number": "EP-560", "title": "Erdős Problem #560", "statement": "Let $\\hat{R}(G)$ denote the size Ramsey number, the minimal number of edges $m$ such that there is a graph $H$ with $m$ edges such that in any $2$-colouring of the edges of $H$ there is a monochromatic copy of $G$.\nDetermine $ \\hat{R}(K_{n,n}), $ where $K_{n,n}$ is the complete bipartite graph with $n$ vertices in each component.", "background": "We know that $ \\frac{1}{60}n^22^n<\\hat{R}(K_{n,n})< \\frac{3}{2}n^32^n. $ The lower bound (which holds for $n\\geq 6$) was proved by Erdos and Rousseau \\cite{ErRo93}. The upper bound was proved by Erdos, Faudree, Rousseau, and Schelp \\cite{EFRS78b} and Ne\\v{s}et\\v{r}il and R\"{o}dl \\cite{NeRo78}.\nConlon, Fox, and Wigderson \\cite{CFW23} have proved that, for any $s\\leq t$, $ \\hat{R}(K_{s,t})\\gg s^{2-\\frac{s}{t}}t2^s, $ and prove that when $t\\gg s\\log s$ we have $\\hat{R}(K_{s,t})\\asymp s^2t2^s$. They conjecture that this should hold for all $s\\leq t$, and so in particular we should have $\\hat{R}(K_{n,n})\\asymp n^32^n$.\nThis problem is #29 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[CFW23] Conlon, David and Fox, Jacob and Wigderson, Yuval, Three early problems on size Ramsey numbers. Combinatorica (2023), 743-768.\n\n[EFRS78b] Erdos, P. and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., The size Ramsey number. Period. Math. Hungar. (1978), 145-161.\n\n[ErRo93] Erdos, P. and Rousseau, C. C., The size Ramsey number of a complete bipartite graph. Discrete Math. (1993), 259-262.\n\n[NeRo78] Ne\\vSet\\v{r}il, J. and R\"{o}dl, V., The structure of critical Ramsey graphs. Acta Math. Acad. Sci. Hungar. (1978), 295-300.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2178, "problem_number": "EP-561", "title": "Erdős Problem #561", "statement": "Let $\\hat{R}(G)$ denote the size Ramsey number, the minimal number of edges $m$ such that there is a graph $H$ with $m$ edges such that in any $2$-colouring of the edges of $H$ there is a monochromatic copy of $G$.\nLet $F_1$ and $F_2$ be the union of stars. More precisely, let $F_1=\\cup_{i\\leq s} K_{1,n_i}$ and $F_2=\\cup_{j\\leq t} K_{1,m_j}$. Prove that $ \\hat{R}(F_1,F_2) = \\sum_{2\\leq k\\leq s+2}\\max\\{n_i+m_j-1 : i+j=k\\}. $ ", "background": "Burr, Erdos, Faudree, Rousseau, and Schelp \\cite{BEFRS78} proved this when all the $n_i$ are identical and all the $m_i$ are identical.\nThis problem is #30 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[BEFRS78] Burr, S. A. and Erdos, P. and Faudree, R. J. and Rousseau,\nC. C. and Schelp, R. H., Ramsey-minimal graphs for multiple copies. Nederl. Akad. Wetensch. Indag. Math. (1978), 187-195.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2179, "problem_number": "EP-562", "title": "Erdős Problem #562", "statement": "Let $R_r(n)$ denote the $r$-uniform hypergraph Ramsey number: the minimal $m$ such that if we $2$-colour all edges of the complete $r$-uniform hypergraph on $m$ vertices then there must be some monochromatic copy of the complete $r$-uniform hypergraph on $n$ vertices.\nProve that, for $r\\geq 3$, $ \\log_{r-1} R_r(n) \\asymp_r n, $ where $\\log_{r-1}$ denotes the $(r-1)$-fold iterated logarithm. That is, does $R_r(n)$ grow like $ 2^{2^{\\cdots n}} $ where the tower of exponentials has height $r-1$?", "background": "A problem of Erdos, Hajnal, and Rado \\cite{EHR65}. A generalisation of [564].\nThis problem is #38 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[EHR65] Erdos, P. and Hajnal, A. and Rado, R., Partition relations for cardinal numbers. Acta Math. Acad. Sci. Hungar. (1965), 93-196.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2180, "problem_number": "EP-563", "title": "Erdős Problem #563", "statement": "Let $F(n,\\alpha)$ denote the smallest $m$ such that there exists a $2$-colouring of the edges of $K_n$ so that every $X\\subseteq [n]$ with $\\lvert X\\rvert\\geq m$ contains more than $\\alpha \\binom{\\lvert X\\rvert}{2}$ many edges of each colour.\nProve that, for every $0\\leq \\alpha< 1/2$, $ F(n,\\alpha)\\sim c_\\alpha\\log n $ for some constant $c_\\alpha$ depending only on $\\alpha$.", "background": "It is easy to show via the probabilistic method that, for every $0\\leq \\alpha<1/2$, $ F(n,\\alpha)\\asymp_\\alpha \\log n. $ Note that when $\\alpha=0$ this is just asking for a $2$-colouring of the edges of $K_n$ which contains no monochromatic clique of size $m$, and hence we recover the classical Ramsey numbers.\nSee also [161] for a generalisation to hypergraphs.\nThis problem is #39 in Ramsey Theory in the graphs problem collection.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2181, "problem_number": "EP-564", "title": "Erdős Problem #564", "statement": "Let $R_3(n)$ be the minimal $m$ such that if the edges of the $3$-uniform hypergraph on $m$ vertices are $2$-coloured then there is a monochromatic copy of the complete $3$-uniform hypergraph on $n$ vertices.\nIs there some constant $c>0$ such that $ R_3(n) \\geq 2^{2^{cn}}? $ ", "background": "A special case of [562]. A problem of Erdos, Hajnal, and Rado \\cite{EHR65}, who prove the bounds $ 2^{cn^2}< R_3(n)< 2^{2^{n}} $ for some constant $c>0$.\nErdos, Hajnal, M\\'{a}t\\'{e}, and Rado \\cite{EHMR84} have proved a doubly exponential lower bound for the corresponding problem with $4$ colours.\nThis problem is #37 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[EHMR84] Erdos, Paul and Hajnal, Andr\\'{a}s and M\\'{a}t\\'{e}, Attila and Rado, Richard, Combinatorial set theory: partition relations for cardinals. (1984), 347.\n\n[EHR65] Erdos, P. and Hajnal, A. and Rado, R., Partition relations for cardinal numbers. Acta Math. Acad. Sci. Hungar. (1965), 93-196.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2182, "problem_number": "EP-566", "title": "Erdős Problem #566", "statement": "Let $G$ be such that any subgraph on $k$ vertices has at most $2k-3$ edges. Is it true that, if $H$ has $m$ edges and no isolated vertices, then $ R(G,H)\\ll m? $ ", "background": "In other words, is $G$ Ramsey size linear? This fails for a graph $G$ with $n$ vertices and $2n-2$ edges (for example with $H=K_n$). Erdos, Faudree, Rousseau, and Schelp \\cite{EFRS93} have shown that any graph $G$ with $n$ vertices and at most $n+1$ edges is Ramsey size linear.\nImplies [567].\nThis problem is #31 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[EFRS93] Erdos, Paul and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., Ramsey size linear graphs. Combin. Probab. Comput. (1993), 389-399.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2183, "problem_number": "EP-567", "title": "Erdős Problem #567", "statement": "Let $G$ be either $Q_3$ or $K_{3,3}$ or $H_5$ (the last formed by adding two vertex-disjoint chords to $C_5$). Is it true that, if $H$ has $m$ edges and no isolated vertices, then $ R(G,H)\\ll m? $ ", "background": "In other words, is $G$ Ramsey size linear? A special case of [566]. In \\cite{Er95} Erdos specifically asks about the case $G=K_{3,3}$.\nThe graph $H_5$ can also be described as $K_4^*$, obtained from $K_4$ by subdividing one edge. ($K_4$ itself is not Ramsey size linear, since $R(4,n)\\gg n^{3-o(1)}$, see [166].) Brada\\'{c}, Gishboliner, and Sudakov \\cite{BGS23} have shown that every subdivision of $K_4$ on at least $6$ vertices is Ramsey size linear, and also that $R(H_5,H) \\ll m$ whenever $H$ is a bipartite graph with $m$ edges and no isolated vertices.\nThis problem is #32 in Ramsey Theory in the graphs problem collection.\nReferences\n\n\n[BGS23] Brada\\'C, D. and Gishboliner, L. and Sudakov, B., On Ramsey size-linear graphs and related questions. arXiv:2202.10388 (2023).\n\n[Er95] Erdos, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2184, "problem_number": "EP-568", "title": "Erdős Problem #568", "statement": "Let $G$ be a graph such that $R(G,T_n)\\ll n$ for any tree $T_n$ on $n$ vertices and $R(G,K_n)\\ll n^2$. Is it true that, for any $H$ with $m$ edges and no isolated vertices, $ R(G,H)\\ll m? $ ", "background": "In other words, is $G$ Ramsey size linear?\nThis problem is #33 in Ramsey Theory in the graphs problem collection.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2185, "problem_number": "EP-569", "title": "Erdős Problem #569", "statement": "Let $k\\geq 1$. What is the best possible $c_k$ such that $ R(C_{2k+1},H)\\leq c_k m $ for any graph $H$ on $m$ edges without isolated vertices?", "background": "This problem is #34 in Ramsey Theory in the graphs problem collection.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2186, "problem_number": "EP-571", "title": "Erdős Problem #571", "statement": "Show that for any rational $\\alpha \\in [1,2)$ there exists a bipartite graph $G$ such that $ \\mathrm{ex}(n;G)\\asymp n^{\\alpha}. $ ", "background": "A problem of Erdos and Simonovits.\nBukh and Conlon \\cite{BuCo18} proved that this holds if we weaken asking for the extremal number of a single graph to asking for the extremal number of a finite family of graphs.\nA rational $\\alpha\\in [1,2)$ for which this holds is known as a Tur\\'{a}n exponent. Known Tur\\'{a}n exponents are:\n{UL}\n{LI} $\\frac{3}{2}-\\frac{1}{2s}$ for $s\\geq 2$ (Conlon, Janzer, and Lee \\cite{CJL21}).{/LI}\n{LI} $\\frac{4}{3}-\\frac{1}{3s}$ and $\\frac{5}{4}-\\frac{1}{4s}$ for $s\\geq 2$ (Jiang and Qiu \\cite{JiQi20}).{/LI}\n{LI} $2-\\frac{a}{b}$ for $\\lfloor b/a\\rfloor^3 \\leq a\\leq \\frac{b}{\\lfloor b/a\\rfloor+1}+1$ (Jiang, Jiang, and Ma \\cite{JJM20}).{/LI}\n{LI} $2-\\frac{a}{b}$ with $b>a\\geq 1$ and $b\\equiv \\pm 1\\pmod{a}$ (Kang, Kim, and Liu \\cite{KKL21}).{/LI}\n{LI} $1+a/b$ with $b>a^2$ (Jiang and Qiu \\cite{JiQi23}),{/LI}\n{LI} $2-\\frac{2}{2b+1}$ for $b\\geq 2$ or $7/5$ (Jiang, Ma, and Yepremyan \\cite{JMY22}).{/LI}\n{LI} $2-a/b$ with $b\\geq (a-1)^2$ (Conlon and Janzer \\cite{CoJa22}).{/LI}\n{/UL}\nSee also [713].\nThis problem is #45 in Extremal Graph Theory in the graphs problem collection.\nReferences\n\n\n[BuCo18] Bukh, Boris and Conlon, David, Rational exponents in extremal graph theory. J. Eur. Math. Soc. (JEMS) (2018), 1747-1757.\n\n[CJL21] Conlon, David and Janzer, Oliver and Lee, Joonkyung, More on the extremal number of subdivisions. Combinatorica (2021), 465-494.\n\n[CoJa22] Conlon, David and Janzer, Oliver, Rational exponents near two. Adv. Comb. (2022), Paper No. 9, 10.\n\n[JJM20] Jiang, Tao and Jiang, Zilin and Ma, Jie, Negligible obstructions and Tur\\'{a}n exponents. arXiv:2007.02975 (2020).\n\n[JMY22] Jiang, Tao and Ma, Jie and Yepremyan, Liana, On Tur\\'{a}n exponents of bipartite graphs. Combin. Probab. Comput. (2022), 333-344.\n\n[JiQi20] Jiang, Tao and Qiu, Yu, Tur\\'{a}n numbers of bipartite subdivisions. SIAM J. Discrete Math. (2020), 556-570.\n\n[JiQi23] Jiang, Tao and Qiu, Yu, Many Tur\\'{a}n exponents via subdivisions. Combin. Probab. Comput. (2023), 134-150.\n\n[KKL21] Kang, Dong Yeap and Kim, Jaehoon and Liu, Hong, On the rational Tur\\'{a}n exponents conjecture. J. Combin. Theory Ser. B (2021), 149-172.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2187, "problem_number": "EP-572", "title": "Erdős Problem #572", "statement": "Show that for $k\\geq 3$ $ \\mathrm{ex}(n;C_{2k})\\gg n^{1+\\frac{1}{k}}. $ ", "background": "It is easy to see that $\\mathrm{ex}(n;C_{2k+1})=\\lfloor n^2/4\\rfloor$ for any $k\\geq 1$ (and $n>2k+1$) (since no bipartite graph contains an odd cycle). Erdos and Klein \\cite{Er38} proved $\\mathrm{ex}(n;C_4)\\asymp n^{3/2}$.\nErdos \\cite{Er64c} and Bondy and Simonovits \\cite{BoSi74} showed that $ \\mathrm{ex}(n;C_{2k})\\ll kn^{1+\\frac{1}{k}}. $ Benson \\cite{Be66} has proved this conjecture for $k=3$ and $k=5$. Lazebnik, Ustimenko, and Woldar \\cite{LUW95} have shown that, for arbitrary $k\\geq 3$, $ \\mathrm{ex}(n;C_{2k})\\gg n^{1+\\frac{2}{3k-3+\nu}}, $ where $\nu=0$ if $k$ is odd and $\nu=1$ if $k$ is even. See \\cite{LUW99} for further history and references.\nSee also [765].\nThis problem is #46 in Extremal Graph Theory in the graphs problem collection.\nReferences\n\n\n[Be66] Benson, Clark T., Minimal regular graphs of girths eight and twelve. Canadian J. Math. (1966), 1091-1094.\n\n[BoSi74] Bondy, J. A. and Simonovits, M., Cycles of even length in graphs. J. Combinatorial Theory Ser. B (1974), 97-105.\n\n[Er38] P. Erdos, On sequences of integers no one of which divides the product of two others and on related problems. Tomsk. Gos. Univ. Ucen Zap. (1938), 74-82.\n\n[Er64c] Erdos, P., Extremal problems in graph theory. Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963) (1964), 29-36.\n\n[LUW95] Lazebnik, F. and Ustimenko, V. A. and Woldar, A. J., A new series of dense graphs of high girth. Bull. Amer. Math. Soc. (N.S.) (1995), 73-79.\n\n[LUW99] Lazebnik, Felix and Ustimenko, Vasiliy A. and Woldar, Andrew\nJ., Polarities and $2k$-cycle-free graphs. Discrete Math. (1999), 503-513.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2188, "problem_number": "EP-573", "title": "Erdős Problem #573", "statement": "Is it true that $ \\mathrm{ex}(n;\\{C_3,C_4\\})\\sim (n/2)^{3/2}? $ ", "background": "A problem of Erdos and Simonovits, who proved that $ \\mathrm{ex}(n;\\{C_4,C_5\\})=(n/2)^{3/2}+O(n). $ K\"{o}v\\'{a}ri, S\\'{o}s, and Tur\\'{a}n \\cite{KST54} proved that the extremal number of edges for containing either $C_4$ or an odd cycle of any length is $\\sim (n/2)^{3/2}$. This problem is therefore asking whether the threshold is the same if we just forbid odd cycles of length $3$.\nSee also [574] for the general case, and [765] for $\\mathrm{ex}(n;C_4)$.\nThis problem is #48 in Extremal Graph Theory in the graphs problem collection.\nReferences\n\n\n[KST54] K\"{o}vari, T. and S\\'{o}s, V. T. and Tur\\'{a}n, P., On a problem of K. Zarankiewicz. Colloq. Math. (1954), 50-57.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2189, "problem_number": "EP-574", "title": "Erdős Problem #574", "statement": "Is it true that, for $k\\geq 2$, $ \\mathrm{ex}(n;\\{C_{2k-1},C_{2k}\\})=(1+o(1))(n/2)^{1+\\frac{1}{k}}. $ ", "background": "A problem of Erdos and Simonovits.\nSee also [573] for the specific case of $k=2$.\nThis problem is #49 in Extremal Graph Theory in the graphs problem collection.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2190, "problem_number": "EP-575", "title": "Erdős Problem #575", "statement": "If $\\mathcal{F}$ is a finite set of finite graphs then $\\mathrm{ex}(n;\\mathcal{F})$ is the maximum number of edges a graph on $n$ vertices can have without containing any subgraphs from $\\mathcal{F}$. Note that it is trivial that $\\mathrm{ex}(n;\\mathcal{F})\\leq \\mathrm{ex}(n;G)$ for every $G\\in\\mathcal{F}$.\nIs it true that, for every $\\mathcal{F}$, if there is a bipartite graph in $\\mathcal{F}$ then there exists some bipartite $G\\in\\mathcal{F}$ such that $ \\mathrm{ex}(n;G)\\ll_{\\mathcal{F}}\\mathrm{ex}(n;\\mathcal{F})? $ ", "background": "A problem of Erdos and Simonovits.\nSee also [180].\nThis problem is #51 in Extremal Graph Theory in the graphs problem collection.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2191, "problem_number": "EP-576", "title": "Erdős Problem #576", "statement": "Let $Q_k$ be the $k$-dimensional hypercube graph (so that $Q_k$ has $2^k$ vertices and $k2^{k-1}$ edges). Determine the behaviour of $ \\mathrm{ex}(n;Q_k). $ ", "background": "Erdos and Simonovits \\cite{ErSi70} proved that $ (\\tfrac{1}{2}+o(1))n^{3/2}\\leq \\mathrm{ex}(n;Q_3) \\ll n^{8/5}. $ (In \\cite{ErSi70} they mention that Erdos had originally conjectured that $ \\mathrm{ex}(n;Q_3)\\gg n^{5/3}$.) Erdos and Simonovits also proved that, if $G$ is the graph $Q_3$ with a missing edge, then $\\mathrm{ex}(n;G)\\asymp n^{3/2}$.\nIn \\cite{Er74c}, \\cite{Er81}, and \\cite{Er93} Erdos asked whether it is $\\mathrm{ex}(n;Q_3)\\asymp n^{8/5}$.\nA theorem of Sudakov and Tomon \\cite{SuTo22} implies $ \\mathrm{ex}(n;Q_k)=o(n^{2-\\frac{1}{k}}). $ Janzer and Sudakov \\cite{JaSu22} have improved this to $ \\mathrm{ex}(n;Q_k)\\ll_k n^{2-\\frac{1}{k-1}+\\frac{1}{(k-1)2^{k-1}}}. $ See also [1035].\nThis problem is #52 in Extremal Graph Theory in the graphs problem collection.\nReferences\n\n\n[Er74c] Erdos, Paul, Extremal problems on graphs and hypergraphs. (1974), 75-84.\n\n[Er81] Erdos, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.\n\n[Er93] Erdos, Paul, Some of my favorite solved and unsolved problems in graph\ntheory. Quaestiones Math. (1993), 333-350.\n\n[ErSi70] Erdos, P. and Simonovits, M., Some extremal problems in graph theory. Combinatorial theory and its applications, I-III (Proc. Colloq., Balatonf\"{u}red, 1969) (1970), 377-390.\n\n[JaSu22] Janzer, O. and Sudakov, B., On the Tur\\'{a}n number of the hypercube. arXiv:2211.02015 (2024).\n\n[SuTo22] Sudakov, Benny and Tomon, Istv\\'{a}n, The extremal number of tight cycles. Int. Math. Res. Not. IMRN (2022), 9663-9684.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2192, "problem_number": "EP-579", "title": "Erdős Problem #579", "statement": "Let $\\delta>0$. If $n$ is sufficiently large and $G$ is a graph on $n$ vertices with no $K_{2,2,2}$ and at least $\\delta n^2$ edges then $G$ contains an independent set of size $\\gg_\\delta n$.", "background": "A problem of Erdos, Hajnal, S\\'{o}s, and Szemer\\'{e}di, who could prove this is true for $\\delta>1/8$.\nSee also [533] and the entry in the graphs problem collection.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2193, "problem_number": "EP-584", "title": "Erdős Problem #584", "statement": "Let $G$ be a graph with $n$ vertices and $\\delta n^{2}$ edges. Are there subgraphs $H_1,H_2\\subseteq G$ such that\n{UL}\n{LI}$H_1$ has $\\gg \\delta^3n^2$ edges and every two edges in $H_1$ are contained in a cycle of length at most $6$, and furthermore if two edges share a vertex they are on a cycle of length $4$, and\n{LI}$H_2$ has $\\gg \\delta^2n^2$ edges and every two edges in $H_2$ are contained in a cycle of length at most $8$.\n{/UL}", "background": "A problem of Erdos, Duke, and R\"{o}dl. Duke and Erdos \\cite{DuEr83}, who proved the first if $n$ is sufficiently large depending on $\\delta$. The real challenge is to prove this when $\\delta=n^{-c}$ for some $c>0$. Duke, Erdos, and R\"{o}dl \\cite{DER84} proved the first statement with a $\\delta^5$ in place of a $\\delta^3$.\nFox and Sudakov \\cite{FoSu08b} have proved the second statement when $\\delta >n^{-1/5}$.\nSee also the entry in the graphs problem collection.\nReferences\n\n\n[DER84] Duke, Richard and Erdos, Paul and R\"{o}dl, Vojt\\vEch, More results on subgraphs with many short cycles. Proceedings of the fifteenth Southeastern conference on\ncombinatorics, graph theory and computing (Baton Rouge,\nLa., 1984) (1984), 295-300.\n\n[DuEr83] No reference found.\n\n\n[FoSu08b] Fox, Jacob and Sudakov, Benny, On a problem of Duke-Erdos-R\"{o}dl on cycle-connected subgraphs. J. Combin. Theory Ser. B (2008), 1056-1062.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2194, "problem_number": "EP-585", "title": "Erdős Problem #585", "statement": "What is the maximum number of edges that a graph on $n$ vertices can have if it does not contain two edge-disjoint cycles with the same vertex set?", "background": "Pyber, R\"{o}dl, and Szemer\\'{e}di \\cite{PRS95} constructed such a graph with $\\gg n\\log\\log n$ edges.\nChakraborti, Janzer, Methuku, and Montgomery \\cite{CJMM24} have shown that such a graph can have at most $n(\\log n)^{O(1)}$ many edges. Indeed, they prove that there exists a constant $C>0$ such that for any $k\\geq 2$ there is a $c_k$ such that if a graph has $n$ vertices and at least $c_kn(\\log n)^{C}$ many edges then it contains $k$ pairwise edge-disjoint cycles with the same vertex set.\nReferences\n\n\n[CJMM24] Chakraborti, D. and Janzer, O. and Methuku, A. and Montgomery, R., Edge-disjoint cycles with the same vertex set. arXiv:2404.07190 (2024).\n\n[PRS95] Pyber, L. and R\"{o}dl, V. and Szemer\\'{e}di, E., Dense subgraphs without 3-regular subgraphs. Journal of Combinatorial Theory, Series B (1995), 41-54.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2195, "problem_number": "EP-588", "title": "Erdős Problem #588", "statement": "Let $f_k(n)$ be minimal such that if $n$ points in $\\mathbb{R}^2$ have no $k+1$ points on a line then there must be at most $f_k(n)$ many lines containing at least $k$ points. Is it true that $ f_k(n)=o(n^2) $ for $k\\geq 4$?", "background": "A generalisation of [101] (which asks about $k=4$).\nThe restriction to $k\\geq 4$ is necessary since Sylvester has shown that $f_3(n)= n^2/6+O(n)$. (See also Burr, Gr\"{u}nbaum, and Sloane \\cite{BGS74} and F\"{u}redi and Pal\\'{a}sti \\cite{FuPa84} for constructions which show that $f_3(n)\\geq(1/6+o(1))n^2$.)\nFor $k\\geq 4$, K\\'{a}rteszi \\cite{Ka63} proved $ f_k(n)\\gg_k n\\log n $ (resolving a conjecture of Erdos that $f_k(n)/n\\to \\infty$). Gr\"{u}nbaum \\cite{Gr76} proved $ f_k(n) \\gg_k n^{1+\\frac{1}{k-2}}. $ Erdos speculated this may be the correct order of magnitude, but Solymosi and Stojakovi\\'{c} \\cite{SoSt13} give a construction which shows $ f_k(n)\\gg_k n^{2-O_k(1/\\sqrt{\\log n})} $ \nReferences\n\n\n[BGS74] Burr, Stefan A. and Gr\"{u}nbaum, Branko and Sloane, N. J. A., The orchard problem. Geometriae Dedicata (1974), 397-424.\n\n[FuPa84] F\"{u}redi, Z. and Pal\\'{a}sti, I., Arrangements of lines with a large number of triangles. Proc. Amer. Math. Soc. (1984), 561-566.\n\n[Gr76] Gr\"{u}nbaum, Branko, New views on some old questions of combinatorial geometry. Colloquio Internazionale sulle Teorie Combinatorie\n(Roma, 1973), Tomo I (1976), 451-468.\n\n[Ka63] F. K\\'{a}rteszi, Sylvester egy t\\'{e}tel\\'{e}r\\H{o}l \\'{e}s Erdos egy sejt\\'{e}s\\'{e}r\\H{o}l. Matematikai Lapok (1963), 3-10.\n\n[SoSt13] Solymosi, J\\'{o}zsef and Stojakovi\\'C, Milo\\vS, Many collinear {$k$}-tuples with no {$k+1$} collinear points. Discrete Comput. Geom. (2013), 811-820.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2196, "problem_number": "EP-589", "title": "Erdős Problem #589", "statement": "Let $g(n)$ be maximal such that in any set of $n$ points in $\\mathbb{R}^2$ with no four points on a line there exists a subset on $g(n)$ points with no three points on a line. Estimate $g(n)$.", "background": "The trivial greedy algorithm gives $g(n)\\gg n^{1/2}$. A similar question can be asked for a set with no $k$ points on a line, searching for a subset with no $l$ points on a line, for any $3\\leq l\\aleph_0$.", "background": "Similar problems were investigated by Erdos, Galvin, and Hajnal \\cite{EGH75}. Erdos claims that for graphs the problem is completely solved: a graph of chromatic number $\\geq \\aleph_1$ must contain all finite bipartite graphs but need not contain any fixed odd cycle.\nReferences\n\n\n[EGH75] Erdos, P. and Galvin, F. and Hajnal, A., On set-systems having large chromatic number and not containing prescribed subsystems. (1975), 425--513.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2200, "problem_number": "EP-595", "title": "Erdős Problem #595", "statement": "Is there an infinite graph $G$ which contains no $K_4$ and is not the union of countably many triangle-free graphs?", "background": "A problem of Erdos and Hajnal. Folkman \\cite{Fo70} and Ne\\v{s}et\\v{r}il and R\"{o}dl \\cite{NeRo75} have proved that for every $n\\geq 1$ there is a graph $G$ which contains no $K_4$ and is not the union of $n$ triangle-free graphs.\nSee also [582] and [596].\nReferences\n\n\n[Fo70] Folkman, Jon, Graphs with monochromatic complete subgraphs in every edge\ncoloring. SIAM J. Appl. Math. (1970), 19-24.\n\n[NeRo75] Ne\\u set\\u ril, Jaroslav and R\"odl, Vojt\\v ech, Type theory of partition properties of graphs. (1975), 405-412.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2201, "problem_number": "EP-596", "title": "Erdős Problem #596", "statement": "For which graphs $G_1,G_2$ is it true that\n{UL}\n{LI} for every $n\\geq 1$ there is a graph $H$ without a $G_1$ but if the edges of $H$ are $n$-coloured then there is a monochromatic copy of $G_2$, and yet{/LI}\n{LI} for every graph $H$ without a $G_1$ there is an $\\aleph_0$-colouring of the edges of $H$ without a monochromatic $G_2$.\n{/UL}", "background": "Erdos and Hajnal originally conjectured that there are no such $G_1,G_2$, but in fact $G_1=C_4$ and $G_2=C_6$ is an example. Indeed, for this pair Ne\\v{s}et\\v{r}il and R\"{o}dl established the first property and Erdos and Hajnal the second (in fact every $C_4$-free graph is a countable union of trees).\nWhether this is true for $G_1=K_4$ and $G_2=K_3$ is the content of [595].\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2202, "problem_number": "EP-597", "title": "Erdős Problem #597", "statement": "Let $G$ be a graph on at most $\\aleph_1$ vertices which contains no $K_4$ and no $K_{\\aleph_0,\\aleph_0}$ (the complete bipartite graph with $\\aleph_0$ vertices in each class). Is it true that $ \\omega_1^2 \\to (\\omega_1\\omega, G)^2? $ What about finite $G$?", "background": "Erdos and Hajnal proved that $\\omega_1^2 \\to (\\omega_1\\omega,3)^2$. Erdos originally asked this with just the assumption that $G$ is $K_4$-free, but Baumgartner proved that $\\omega_1^2 \not\\to (\\omega_1\\omega, K_{\\aleph_0,\\aleph_0})^2$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2203, "problem_number": "EP-598", "title": "Erdős Problem #598", "statement": "Let $m$ be an infinite cardinal and $\\kappa$ be the successor cardinal of $2^{\\aleph_0}$. Can one colour the countable subsets of $m$ using $\\kappa$ many colours so that every $X\\subseteq m$ with $\\lvert X\\rvert=\\kappa$ contains subsets of all possible colours?\n\",\n \"difficulty\": \"L1\"\n},{", "background": "", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2204, "problem_number": "EP-600", "title": "Erdős Problem #600", "statement": "Let $e(n,r)$ be minimal such that every graph on $n$ vertices with at least $e(n,r)$ edges, each edge contained in at least one triangle, must have an edge contained in at least $r$ triangles. Let $r\\geq 2$. Is it true that $ e(n,r+1)-e(n,r)\\to \\infty $ as $n\\to \\infty$? Is it true that $ \\frac{e(n,r+1)}{e(n,r)}\\to 1 $ as $n\\to \\infty$?", "background": "Ruzsa and Szemer\\'{e}di \\cite{RuSz78} proved that $e(n,r)=o(n^2)$ for any fixed $r$.\nSee also [80].\nReferences\n\n\n[RuSz78] Ruzsa, I. Z. and Szemer\\'{e}di, E., Triple systems with no six points carrying three triangles. Combinatorics (Proc. Fifth Hungarian Colloq.,\nKeszthely, 1976), Vol. II (1978), 939-945.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2205, "problem_number": "EP-601", "title": "Erdős Problem #601", "statement": "For which limit ordinals $\\alpha$ is it true that if $G$ is a graph with vertex set $\\alpha$ then $G$ must have either an infinite path or independent set on a set of vertices with order type $\\alpha$?", "background": "A problem of Erdos, Hajnal, and Milner \\cite{EHM70}, who proved this is true for $\\alpha < \\omega_1^{\\omega+2}$.\nIn \\cite{Er82e} Erdos offers \\$250 for showing what happens when $\\alpha=\\omega_1^{\\omega+2}$ and \\$500 for settling the general case.\nLarson \\cite{La90} proved this is true for all $\\alpha<2^{\\aleph_0}$ assuming Martin's axiom.\nReferences\n\n\n[EHM70] Erdos, P. and Hajnal, A. and Milner, E. C., Set mappings and polarized partition relations. Combinatorial theory and its applications, I-III (Proc.\nColloq., Balatonf\"{u}red, 1969) (1970), 327-363.\n\n[Er82e] Erdos, Paul, Some of my favourite problems which recently have been solved. (1982), 59--79.\n\n[La90] Larson, Jean A., Martin's axiom and ordinal graphs: large independent sets or infinite paths. Ann. Pure Appl. Logic (1990), 31-39.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2206, "problem_number": "EP-602", "title": "Erdős Problem #602", "statement": "Let $(A_i)$ be a family of sets with $\\lvert A_i\\rvert=\\aleph_0$ for all $i$, such that for any $i\neq j$ we have $\\lvert A_i\\cap A_j\\rvert$ finite and $\neq 1$. Is there a $2$-colouring of $\\cup A_i$ such that no $A_i$ is monochromatic?", "background": "A problem of Komj\\'{a}th. The existence of such a $2$-colouring is sometimes known as Property B.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2207, "problem_number": "EP-603", "title": "Erdős Problem #603", "statement": "Let $(A_i)$ be a family of countably infinite sets such that $\\lvert A_i\\cap A_j\\rvert \neq 2$ for all $i\neq j$. Find the smallest cardinal $C$ such that $\\cup A_i$ can always be coloured with at most $C$ colours so that no $A_i$ is monochromatic.", "background": "A problem of Komj\\'{a}th. If instead we have $\\lvert A_i\\cap A_j\\rvert \neq 1$ then Komj\\'{a}th showed that this is possible with at most $\\aleph_0$ colours.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2208, "problem_number": "EP-604", "title": "Erdős Problem #604", "statement": "Given $n$ distinct points $A\\subset\\mathbb{R}^2$ must there be a point $x\\in A$ such that $ \\#\\{ d(x,y) : y \\in A\\} \\gg n^{1-o(1)}? $ Or even $\\gg n/\\sqrt{\\log n}$?", "background": "The pinned distance problem, a stronger form of [89]. The example of an integer grid show that $n/\\sqrt{\\log n}$ would be best possible.\nIt may be true that there are $\\gg n$ many such points, or that this is true on average - for example, if $d(x)$ counts the number of distinct distances from $x$ then in \\cite{Er75f} Erdos conjectured $ \\sum_{x\\in A}d(x) \\gg \\frac{n^2}{\\sqrt{\\log n}}, $ where $A\\subset \\mathbb{R}^2$ is any set of $n$ points.\nIn \\cite{Er97e} Erdos offers \\$500 for a solution to this problem, but it is unclear whether he intended this for proving the existence of a single such point or for $\\gg n$ many such points.\nIn \\cite{Er97e} Erdos wrote that he initially 'overconjectured' and thought that the answer to this problem is the same as for the number of distinct distances between all pairs (see [89]), but this was disproved by Harborth. It could be true that the answers are the same up to an additive factor of $n^{o(1)}$.\nThe best known bound is $ \\gg n^{c-o(1)}, $ due to Katz and Tardos \\cite{KaTa04}, where $ c=\\frac{48-14e}{55-16e}=0.864137\\cdots. $ \nReferences\n\n\n[Er75f] Erdos, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108.\n\n[Er97e] Erdos, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537.\n\n[KaTa04] Katz, Nets Hawk and Tardos, G\\'{a}bor, A new entropy inequality for the Erdos distance problem. Towards a theory of geometric graphs (2004), 119-126.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2209, "problem_number": "EP-609", "title": "Erdős Problem #609", "statement": "Let $f(n)$ be the minimal $m$ such that if the edges of $K_{2^n+1}$ are coloured with $n$ colours then there must be a monochromatic odd cycle of length at most $m$. Estimate $f(n)$.", "background": "A problem of Erdos and Graham. The edges of $K_{2^n}$ can be $n$-coloured to avoid odd cycles of any length. It can be shown that $C_5$ and $C_7$ can be avoided for large $n$.\nChung \\cite{Ch97} asked whether $f(n)\\to \\infty$ as $n\\to \\infty$. Day and Johnson \\cite{DaJo17} proved this is true, and that $ f(n)\\geq 2^{c\\sqrt{\\log n}} $ for some constant $c>0$. The trivial upper bound is $2^n$.\nGir\\~{a}o and Hunter \\cite{GiHu24} have proved that $ f(n) \\ll \\frac{2^n}{n^{1-o(1)}}. $ Janzer and Yip \\cite{JaYi25} have improved this to $ f(n) \\ll n^{3/2}2^{n/2}. $ See also the entry in the graphs problem collection.\nReferences\n\n\n[Ch97] Chung, F. R. K., Open problems of {P}aul Erdos in graph theory. J. Graph Theory (1997), 3--36.\n\n[DaJo17] Day, A. Nicholas and Johnson, J. Robert, Multicolour Ramsey numbers of odd cycles. J. Combin. Theory Ser. B (2017), 56-63.\n\n[GiHu24] A. Gir\\~Ao and Z. Hunter, Monochromatic odd cycles in edge-coloured complete graphs. arXiv:2412.07708 (2024).\n\n[JaYi25] O. Janzer and F. Yip, Short monochromatic odd cycles. arXiv:2506.14910 (2025).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2210, "problem_number": "EP-610", "title": "Erdős Problem #610", "statement": "For a graph $G$ let $\\tau(G)$ denote the minimal number of vertices that include at least one from each maximal clique of $G$ (sometimes called the clique transversal number).\nEstimate $\\tau(G)$. In particular, is it true that if $G$ has $n$ vertices then $ \\tau(G) \\leq n-\\omega(n)\\sqrt{n} $ for some $\\omega(n)\\to \\infty$, or even $ \\tau(G) \\leq n-c\\sqrt{n\\log n} $ for some absolute constant $c>0$?", "background": "A problem of Erdos, Gallai, and Tuza \\cite{EGT92}, who proved that $ \\tau(G) \\leq n-\\sqrt{2n}+O(1). $ This would be best possible, since there exist triangle-free graphs with all independent sets of size $O(\\sqrt{n\\log n})$, which follows from the lower bound for $R(3,k)$ by Kim \\cite{Ki95} (see [165]).\nIndeed, Erdos, Gallai, and Tuza speculate that if $f(n)$ is the largest $k$ such that every triangle-free graph on $n$ vertices contains an independent set on $f(n)$ vertices, then $\\tau(G)\\leq n-f(n)$.\nA positive answer to this problem would follow from a positive answer to [151] (since Ajtai, Koml\\'{o}s, and Szemer\\'{e}di \\cite{AKS80} have proved that the $H(n)$ defined there satisfies $H(n)\\gg \\sqrt{n\\log n}$).\nSee also [151], [611], this entry and and this entry in the graphs problem collection.\nReferences\n\n\n[AKS80] Ajtai, Mikl\\'{o}s and Koml\\'{o}s, J\\'{a}nos and Szemer\\'{e}di, Endre, A note on Ramsey numbers. J. Combin. Theory Ser. A (1980), 354-360.\n\n[EGT92] Erdos, Paul and Gallai, Tibor and Tuza, Zsolt, Covering the cliques of a graph with vertices. Discrete Math. (1992), 279-289.\n\n[Ki95] Kim, J. H., The Ramsey number $R(3,t)$ has order of magnitude $t^2/\\log t$. Random Structures and Algorithms (1995), 173-207.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2211, "problem_number": "EP-611", "title": "Erdős Problem #611", "statement": "For a graph $G$ let $\\tau(G)$ denote the minimal number of vertices that include at least one from each maximal clique of $G$ (sometimes called the clique transversal number).\nIs it true that if all maximal cliques in $G$ have at least $cn$ vertices then $\\tau(G)=o_c(n)$?\nSimilarly, estimate for $c>0$ the minimal $k_c(n)$ such that if every maximal clique in $G$ has at least $k_c(n)$ vertices then $\\tau(G)<(1-c)n$.", "background": "A problem of Erdos, Gallai, and Tuza \\cite{EGT92}, who proved for the latter question that $k_c(n) \\geq n^{c'/\\log\\log n}$ for some $c'>0$, and that if every clique has size least $k$ then $\\tau(G) \\leq n-(kn)^{1/2}$. Bollob\\'{a}s and Erdos proved that if every maximal clique has at least $n+3-2\\sqrt{n}$ vertices then $\\tau(G)=1$ (and this threshold is best possible).\nSee also [610] and the entry in the graphs problem collection.\nReferences\n\n\n[EGT92] Erdos, Paul and Gallai, Tibor and Tuza, Zsolt, Covering the cliques of a graph with vertices. Discrete Math. (1992), 279-289.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2212, "problem_number": "EP-612", "title": "Erdős Problem #612", "statement": "Let $G$ be a connected graph with $n$ vertices, minimum degree $d$, and diameter $D$. Show if that $G$ contains no $K_{2r}$ and $(r-1)(3r+2)\\mid d$ then $ D\\leq \\frac{2(r-1)(3r+2)}{2r^2-1}\\frac{n}{d}+O(1), $ and if $G$ contains no $K_{2r+1}$ and $3r-1 \\mid d$ then $ D\\leq \\frac{3r-1}{r}\\frac{n}{d}+O(1). $ ", "background": "A problem of Erdos, Pach, Pollack, and Tuza \\cite{EPPT89}, who gave constructions showing that the above bounds would be sharp, and proved the case $2r+1=3$. It is known (see \\cite{EPPT89} for example) that any connected graph on $n$ vertices with minimum degree $d$ has diameter $ D\\leq 3\\frac{n}{d+1}+O(1). $ This was disproven for the case of $K_{2r}$-free graphs with $r\\geq 2$ by Czabarka, Singgih, and Sz\\'{e}kely \\cite{CSS21}, who constructed arbitrarily large connected graphs on $n$ vertices which contain no $K_{2r}$ and have minimum degree $d$, and diameter $ \\frac{6r-5}{(2r-1)d+2r-3}n+O(1), $ which contradicts the above conjecture for each fixed $r$ as $d\\to \\infty$.\nThey suggest the amended conjecture, which no longer divides into two cases, that if $G$ is a connected graph on $n$ vertices with minimum degree $d$ which contains no $K_{k+1}$ then the diameter of $G$ is at most $ (3-\\tfrac{2}{k})\\frac{n}{d}+O(1). $ This bound is known under the weaker assumption that $G$ is $k$-colourable when $k=3$ and $k=4$, shown by Czabarka, Dankelmann, and Sz\\'{e}kely \\cite{CDS09} and Czabarka, Smith, and Sz\\'{e}kely \\cite{CSS23}.\nCambie and Jooken \\cite{CaJo25} have given an example that shows the diameter for $K_4$-free graphs with minimum degree $16$ is at least $\\frac{31}{216}n+O(1)$, giving another counterexample to the original conjecture.\nSee also the entry in the graphs problem collection.\nReferences\n\n\n[CDS09] Czabarka, \\'{e}. and Dankelmann, P. and Sz\\'{e}kely, L. A., Diameter of 4-colourable graphs. European J. Combin. (2009), 1082--1089.\n\n[CSS21] Czabarka, \\'{e}va and Singgih, Inne and Sz\\'{e}kely, L\\'aszl\\'{o}{}\nA., Counterexamples to a conjecture of {E}rd\\H{o}s, {P}ach,\n{P}ollack and {T}uza. J. Combin. Theory Ser. B (2021), 38--45.\n\n[CSS23] Czabarka, \\'{e}va and Smith, Stephen J. and Sz\\'{e}kely,\nL\\'aszl\\'{o}, Maximum diameter of 3- and 4-colorable graphs. J. Graph Theory (2023), 262--270.\n\n[CaJo25] S. Cambie and J. Jooken, Sharp results for the Erdos, Pach, Pollack and Tuza problem. arXiv:2502.08626 (2025).\n\n[EPPT89] No reference found.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2213, "problem_number": "EP-614", "title": "Erdős Problem #614", "statement": "Let $f(n,k)$ be minimal such that there is a graph with $n$ vertices and $f(n,k)$ edges where every set of $k+2$ vertices induces a subgraph with maximum degree at least $k$. Determine $f(n,k)$.\n\",\n \"difficulty\": \"L1\"\n},{", "background": "", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2214, "problem_number": "EP-616", "title": "Erdős Problem #616", "statement": "Let $r\\geq 3$. For an $r$-uniform hypergraph $G$ let $\\tau(G)$ denote the covering number (or transversal number), the minimum size of a set of vertices which includes at least one from each edge in $G$.\nDetermine the best possible $t$ such that, if $G$ is an $r$-uniform hypergraph $G$ where every subgraph $G'$ on at most $3r-3$ vertices has $\\tau(G')\\leq 1$, we have $\\tau(G)\\leq t$.", "background": "Erdos, Hajnal, and Tuza \\cite{EHT91} proved that this $t$ satisfies $ \\frac{3}{16}r+\\frac{7}{8}\\leq t \\leq \\frac{1}{5}r. $ \nReferences\n\n\n[EHT91] Erdos, Paul and Hajnal, Andr\\'{a}s and Tuza, Zsolt, Local constraints ensuring small representing sets. J. Combin. Theory Ser. A (1991), 78-84.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2215, "problem_number": "EP-619", "title": "Erdős Problem #619", "statement": "For a triangle-free graph $G$ let $h_r(G)$ be the smallest number of edges that need to be added to $G$ so that it has diameter $r$ (while preserving the property of being triangle-free).\nIs it true that there exists a constant $c>0$ such that if $G$ is a connected graph on $n$ vertices then $h_4(G)<(1-c)n$?", "background": "A problem of Erdos, Gy\\'{a}rf\\'{a}s, and Ruszink\\'{o} \\cite{EGR98} who proved that $h_3(G)\\leq n$ and $h_5(G) \\leq \\frac{n-1}{2}$ and there exist connected graphs $G$ on $n$ vertices with $h_3(G)\\geq n-c$ for some constant $c>0$.\nIf we omit the condition that the graph must remain triangle-free then Alon, Gy\\'{a}rf\\'{a}s, and Ruszink\\'{o} \\cite{AGR00} have proved that adding $n/2$ edges always suffices to obtain diameter at most $4$.\nSee also [134] and [618].\nReferences\n\n\n[AGR00] Alon, Noga and Gy\\'{a}rf\\'{a}s, Andr\\'{a}s and Ruszink\\'{o}, Mikl\\'{o}s, Decreasing the diameter of bounded degree graphs. J. Graph Theory (2000), 161--172.\n\n[EGR98] Erdos, Paul and Gy\\'{a}rf\\'{a}s, Andr\\'{a}s and\nRuszink\\'{o}, Mikl\\'{o}s, How to decrease the diameter of triangle-free graphs. Combinatorica (1998), 493-501.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2216, "problem_number": "EP-620", "title": "Erdős Problem #620", "statement": "If $G$ is a graph on $n$ vertices without a $K_4$ then how large a triangle-free induced subgraph must $G$ contain?", "background": "This was first asked by Erdos and Rogers \\cite{ErRo62}, and is generally known as the Erdos-Rogers problem. Let $f(n)$ be such that every such graph contains a triangle-free subgraph with at least $f(n)$ vertices.\nIt is now known that $f(n)=n^{1/2+o(1)}$. Bollob\\'{a}s and Hind \\cite{BoHi91} proved $ n^{1/2} \\ll f(n) \\ll n^{7/10+o(1)}. $ Krivelevich \\cite{Kr94} improved this to $ n^{1/2}(\\log\\log n)^{1/2} \\ll f(n) \\ll n^{2/3}(\\log n)^{1/3}. $ Wolfovitz \\cite{Wo13} proved $ f(n) \\ll n^{1/2}(\\log n)^{120}. $ The best bounds currently known are $ n^{1/2}\\frac{(\\log n)^{1/2}}{\\log\\log n}\\ll f(n) \\ll n^{1/2}\\log n. $ The lower bound follows from results of Shearer \\cite{Sh95}, and the upper bound was proved by Mubayi and Verstraete \\cite{MuVe24}.\nReferences\n\n\n[BoHi91] Bollob\\'{a}s, B. and Hind, H. R., Graphs without large triangle free subgraphs. Discrete Math. (1991), 119-131.\n\n[ErRo62] Erdos, P. and Rogers, C. A., The construction of certain graphs. Canadian J. Math. (1962), 702-707.\n\n[Kr94] Krivelevich, Michael, {$K^s$}-free graphs without large {$K^r$}-free subgraphs. Combin. Probab. Comput. (1994), 349-354.\n\n[MuVe24] D. Mubayi and J. Verstraete, On the order of Erdos-Rogers functions. arXiv:2401.02548 (2024).\n\n[Sh95] Shearer, James B., On the independence number of sparse graphs. Random Structures Algorithms (1995), 269--271.\n\n[Wo13] Wolfovitz, Guy, {$K_4$}-free graphs without large induced triangle-free\nsubgraphs. Combinatorica (2013), 623-631.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2217, "problem_number": "EP-623", "title": "Erdős Problem #623", "statement": "Let $X$ be a set of cardinality $\\aleph_\\omega$ and $f$ be a function from the finite subsets of $X$ to $X$ such that $f(A)\not\\in A$ for all $A$. Must there exist an infinite $Y\\subseteq X$ that is independent - that is, for all finite $B\\subset Y$ we have $f(B)\not\\in Y$?", "background": "A problem of Erdos and Hajnal \\cite{ErHa58}, who proved that if $\\lvert X\\rvert <\\aleph_\\omega$ then the answer is no. Erdos suggests in \\cite{Er99} that this problem is 'perhaps undecidable'.\nReferences\n\n\n[Er99] Erdos, Paul, A selection of problems and results in combinatorics. Combin. Probab. Comput. (1999), 1-6.\n\n[ErHa58] Erdos, P. and Hajnal, A., On the structure of set mappings. Acta Math. Acad. Sci. Hungar. (1958), 111-133.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2218, "problem_number": "EP-624", "title": "Erdős Problem #624", "statement": "Let $X$ be a finite set of size $n$ and $H(n)$ be such that there is a function $f:\\{A : A\\subseteq X\\}\\to X$ so that for every $Y\\subseteq X$ with $\\lvert Y\\rvert \\geq H(n)$ we have $ \\{ f(A) : A\\subseteq Y\\}=X. $ Prove that $ H(n)-\\log_2 n \\to \\infty. $ ", "background": "A problem of Erdos and Hajnal \\cite{ErHa68} who proved that $ \\log_2 n \\leq H(n) < \\log_2n +(3+o(1))\\log_2\\log_2n. $ Erdos said that even the weaker statement that for $n=2^k$ we have $H(n)\\geq k+1$ is open, but Alon has provided the following simple proof: by the pigeonhole principle there are $\\frac{n-1}{2}$ subsets $A_i$ of size $2$ such that $f(A_i)$ is the same. Any set $Y$ of size $k$ containing at least $k/2$ of them can have at most $ 2^k-\\lfloor k/2\\rfloor+1< 2^k=n $ distinct elements in the union of the images of $f(A)$ for $A\\subseteq Y$.\nFor this weaker statement, Erdos and Gy\\'{a}rf\\'{a}s conjectured the stronger form that if $\\lvert X\\rvert=2^k$ then, for any $f:\\{A : A\\subseteq X\\}\\to X$, there must exist some $Y\\subset X$ of size $k$ such that $ \\#\\{ f(A) : A\\subseteq Y\\}< 2^k-k^C $ for every $C$ (with $k$ sufficiently large depending on $C$). This was proved by Alon (personal communication), who proved the stronger version that there exists some absolute constant $c>0$ such that, if $k$ is large enough, there must exist some $Y\\subset X$ of size $k$ such that $ \\#\\{ f(A) : A\\subseteq Y\\}<(1-c)2^k. $ Alon also proved that, provided $k$ is large enough, if $\\lvert X\\rvert=2^k$ there exists some $f:\\{A: A\\subseteq X\\}\\to X$ such that, if $Y\\subset X$ with $\\lvert Y\\rvert=k$, then $ \\#\\{ f(A) : A\\subseteq Y\\}>\\tfrac{1}{4}2^k. $ \nReferences\n\n\n[ErHa68] Erdos, P. and Hajnal, A., On a combinatorial problem. Mat. Lapok (1968), 345-348.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2219, "problem_number": "EP-625", "title": "Erdős Problem #625", "statement": "The cochromatic number of $G$, denoted by $\\zeta(G)$, is the minimum number of colours needed to colour the vertices of $G$ such that each colour class induces either a complete graph or empty graph. Let $\\chi(G)$ denote the chromatic number.\nIf $G$ is a random graph with $n$ vertices and each edge included independently with probability $1/2$ then is it true that almost surely $ \\chi(G) - \\zeta(G) \\to \\infty $ as $n\\to \\infty$?", "background": "A problem of Erdos and Gimbel (see also \\cite{Gi16}). At a conference on random graphs in Poznan, Poland (most likely in 1989) Erdos offered \\$100 for a proof that this is true, and \\$1000 for a proof that this is false (although later told Gimbel that \\$1000 was perhaps too much).\nIt is known that almost surely $ \\frac{n}{2\\log_2n}\\leq \\zeta(G)\\leq \\chi(G)\\leq (1+o(1))\\frac{n}{2\\log_2n}. $ (The final upper bound is due to Bollob\\'{a}s \\cite{Bo88}. The first inequality follows from the fact that almost surely $G$ has clique number and independence number $< 2\\log_2n$.)\nHeckel \\cite{He24} and, independently, Steiner \\cite{St24b} have shown that it is not the case that $\\chi(G)-\\zeta(G)$ is bounded with high probability, and in fact if $\\chi(G)-\\zeta(G) \\leq f(n)$ with high probability then $f(n)\\geq n^{1/2-o(1)}$ along an infinite sequence of $n$. Heckel conjectures that, with high probability, $ \\chi(G)-\\zeta(G) \\asymp \\frac{n}{(\\log n)^3}. $ Heckel \\cite{He24c} further proved that, for any $\\epsilon>0$, we have $ \\chi(G) -\\zeta(G) \\geq n^{1-\\epsilon} $ for roughly $95\\%$ of all $n$.\nReferences\n\n\n[Bo88] Bollob\\'{a}s, B., The chromatic number of random graphs. Combinatorica (1988), 49-55.\n\n[Gi16] J. Gimbel, Some of my favorite coloring problems for graphs and digraphs. Graph Theory: Favorite conjectures and open problems (2016), 95-108.\n\n[He24] A. Heckel, On a question of Erdos and Gimbel on the cochromatic number. arXiv:2408.13839 (2024).\n\n[He24c] A. Heckel, The difference between the chromatic and the cochromatic number of a random graph. arXiv:2409.17614 (2024).\n\n[St24b] R. Steiner, On the difference between the chromatic and cochromatic number. arXiv:2408.02400 (2024).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2220, "problem_number": "EP-626", "title": "Erdős Problem #626", "statement": "Let $k\\geq 4$ and $g_k(n)$ denote the largest $m$ such that there is a graph on $n$ vertices with chromatic number $k$ and girth $>m$ (i.e. contains no cycle of length $\\leq m$). Does $ \\lim_{n\\to \\infty}\\frac{g_k(n)}{\\log n} $ exist?\nConversely, if $h^{(m)}(n)$ is the maximal chromatic number of a graph on $n$ vertices with girth $>m$ then does $ \\lim_{n\\to \\infty}\\frac{\\log h^{(m)}(n)}{\\log n} $ exist, and what is its value?", "background": "It is known that $ \\frac{1}{4\\log k}\\log n\\leq g_k(n) \\leq \\frac{2}{\\log(k-2)}\\log n+1, $ the lower bound due to Kostochka \\cite{Ko88} and the upper bound to Erdos \\cite{Er59b}.\nErdos \\cite{Er59b} proved that $ \\lim_{n\\to \\infty}\\frac{\\log h^{(m)}(n)}{\\log n}\\gg \\frac{1}{m} $ and, for odd $m$, $ \\lim_{n\\to \\infty}\\frac{\\log h^{(m)}(n)}{\\log n}\\leq \\frac{2}{m+1}, $ and conjectured this is sharp. He had no good guess for the value of the limit for even $m$, other that it should lie in $[\\frac{2}{m+2},\\frac{2}{m}]$, but could not prove this even for $m=4$.\nSee also the entry in the graphs problem collection.\nReferences\n\n\n[Er59b] Erdos, P., Graph theory and probability. Canadian J. Math. (1959), 34-38.\n\n[Ko88] Kostochka, A. V., Upper bounds on the chromatic number of graphs. Trudy Inst. Mat. (Novosibirsk) (1988), 204-226, 265.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2221, "problem_number": "EP-627", "title": "Erdős Problem #627", "statement": "Let $\\omega(G)$ denote the clique number of $G$ and $\\chi(G)$ the chromatic number. If $f(n)$ is the maximum value of $\\chi(G)/\\omega(G)$, as $G$ ranges over all graphs on $n$ vertices, then does $ \\lim_{n\\to\\infty}\\frac{f(n)}{n/(\\log n)^2} $ exist?", "background": "Tutte and Zykov \\cite{Zy52} independently proved that for every $k$ there is a graph with $\\omega(G)=2$ and $\\chi(G)=k$. Erdos \\cite{Er61d} proved that for every $n$ there is a graph on $n$ vertices with $\\omega(G)=2$ and $\\chi(G)\\gg n^{1/2}/\\log n$, whence $f(n) \\gg n^{1/2}/\\log n$.\nErdos \\cite{Er67c} proved that $ f(n) \\asymp \\frac{n}{(\\log n)^2} $ and that the limit in question, if it exists, must be in $ (\\log 2)^2\\cdot [1/4,1]. $ See also the entry in the graphs problem collection.\nReferences\n\n\n[Er61d] Erdos, P., Graph theory and probability. II. Canadian J. Math. (1961), 346-352.\n\n[Er67c] Erdos, P., Some remarks on chromatic graphs. Colloq. Math. (1967), 253-256.\n\n[Zy52] Zykov, A. A., On some properties of linear complexes. Amer. Math. Soc. Translation (1952), 33.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2222, "problem_number": "EP-629", "title": "Erdős Problem #629", "statement": "The list chromatic number $\\chi_L(G)$ is defined to be the minimal $k$ such that for any assignment of a list of $k$ colours to each vertex of $G$ (perhaps different lists for different vertices) a colouring of each vertex by a colour on its list can be chosen such that adjacent vertices receive distinct colours.\nDetermine the minimal number of vertices $n(k)$ of a bipartite graph $G$ such that $\\chi_L(G)>k$.", "background": "A problem of Erdos, Rubin, and Taylor \\cite{ERT80}, who proved that $ 2^{k-1}0$: take $A$ to be the union of all odd numbers together with numbers of the shape $2^k$ with $k$ odd.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2226, "problem_number": "EP-638", "title": "Erdős Problem #638", "statement": "Let $S$ be a family of finite graphs such that for every $n$ there is some $G_n\\in S$ such that if the edges of $G_n$ are coloured with $n$ colours then there is a monochromatic triangle.\nIs it true that for every infinite cardinal $\\aleph$ there is a graph $G$ of which every finite subgraph is in $S$ and if the edges of $G$ are coloured with $\\aleph$ many colours then there is a monochromatic triangle.", "background": "Erdos writes 'if the answer is affirmative many extensions and generalisations will be possible'.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2227, "problem_number": "EP-640", "title": "Erdős Problem #640", "statement": "Is there some function $f$ such that for all $k\\geq 3$ if a finite graph $G$ has chromatic number $\\geq f(k)$ then $G$ must contain some odd cycle whose vertices span a graph of chromatic number $\\geq k$?\n\",\n \"difficulty\": \"L1\"\n},{", "background": "", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2228, "problem_number": "EP-642", "title": "Erdős Problem #642", "statement": "Let $f(n)$ be the maximal number of edges in a graph on $n$ vertices such that all cycles have more vertices than diagonals. Is it true that $f(n)\\ll n$?", "background": "A problem of Hamburger and Szegedy.\nChen, Erdos, and Staton \\cite{CES96} proved $f(n) \\ll n^{3/2}$. Dragani\\'{c}, Methuku, Munh\\'{a} Correia, and Sudakov \\cite{DMMS24} have improved this to $ f(n) \\ll n(\\log n)^8. $ \nReferences\n\n\n[CES96] Chen, Guantao and Erdos, Paul and Staton, William, Proof of a conjecture of {B}ollob\\'as on nested cycles. J. Combin. Theory Ser. B (1996), 38--43.\n\n[DMMS24] Dragani\\'c, Nemanja and Methuku, Abhishek and Munh\\'a{}\nCorreia, David and Sudakov, Benny, Cycles with many chords. Random Structures Algorithms (2024), 3--16.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2229, "problem_number": "EP-643", "title": "Erdős Problem #643", "statement": "Let $f(n;t)$ be minimal such that if a $t$-uniform hypergraph on $n$ vertices contains at least $f(n;t)$ edges then there must be four edges $A,B,C,D$ such that $ A\\cup B= C\\cup D $ and $ A\\cap B=C\\cap D=\\emptyset. $ Estimate $f(n;t)$ - in particular, is it true that for $t\\geq 3$ $ f(n;t)=(1+o(1))\\binom{n}{t-1}? $ ", "background": "For $t=2$ this is asking for the maximal number of edges on a graph which contains no $C_4$, and so $f(n;2)=(1/2+o(1))n^{3/2}$.\nF\"{u}redi \\cite{Fu84} proved that $f(n;3) \\ll n^2$ and $f(n;3) > \\binom{n}{2}$ for infinitely many $n$. Pikhurko and Verstra\"{e}te \\cite{PiVe09} have proved $f(n;3)\\leq \\frac{13}{9}\\binom{n}{2}$ for all $n$.\nMore generally, F\"{u}redi \\cite{Fu84} proved that $ \\binom{n-1}{t-1}+\\left\\lfloor\\frac{n-1}{t}\\right\\rfloor\\leq f(n;t) < \\frac{7}{2}\\binom{n}{t-1}, $ and conjectured the lower bound is sharp for $t\\geq 4$. Pikhurko and Verstra\"{e}te \\cite{PiVe09} have proved that $ 1 \\leq \\limsup_{n\\to \\infty} \\frac{f(n;t)}{\\binom{n}{t-1}}\\leq \\min\\left(\\frac{7}{4},1+\\frac{2}{\\sqrt{t}}\\right) $ for all $t\\geq 3$.\nF\"{u}redi \\cite{Fu84} proved that $f(n;3)/\\binom{n}{2}$ converges as $n\\to \\infty$, but the existence of the limit for $t\\geq 4$ is unknown.\nReferences\n\n\n[Fu84] F\"uredi, Z., Hypergraphs in which all disjoint pairs have distinct unions. Combinatorica (1984), 161--168.\n\n[PiVe09] Pikhurko, Oleg and Verstra\"{e}te, Jacques, The maximum size of hypergraphs without generalized 4-cycles. J. Combin. Theory Ser. A (2009), 637--649.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2230, "problem_number": "EP-644", "title": "Erdős Problem #644", "statement": "Let $f(k,r)$ be minimal such that if $A_1,A_2,\\ldots$ is a family of sets, all of size $k$, such that for every collection of $r$ of the $A_is$ there is some pair $\\{x,y\\}$ which intersects all of the $A_j$, then there is some set of size $f(k,r)$ which intersects all of the sets $A_i$. Is it true that $ f(k,7)=(1+o(1))\\frac{3}{4}k? $ Is it true that for any $r\\geq 3$ there exists some constant $c_r$ such that $ f(k,r)=(1+o(1))c_rk? $ ", "background": "A problem of Erdos, Fon-Der-Flaass, Kostochka, and Tuza \\cite{EFKT92}, who proved that $f(k,3)=2k$ and $f(k,4)=\\lfloor 3k/2\\rfloor$ and $f(k,5)=\\lfloor 5k/4\\rfloor$, and further that $f(k,6)=k$.\nReferences\n\n\n[EFKT92] Erd\"{o}s, P. and Fon-Der-Flaass, D. and Kostochka, A. V. and\nTuza, Zs., Small transversals in uniform hypergraphs. Siberian Adv. Math. (1992), 82-88.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2231, "problem_number": "EP-650", "title": "Erdős Problem #650", "statement": "Let $f(m)$ be such that if $A\\subseteq \\{1,\\ldots,N\\}$ has $\\lvert A\\rvert=m$ then every interval in $[1,\\infty)$ of length $2N$ contains $\\geq f(m)$ many distinct integers $b_1,\\ldots,b_r$ where each $b_i$ is divisible by some $a_i\\in A$, where $a_1,\\ldots,a_r$ are distinct.\nEstimate $f(m)$. In particular is it true that $f(m)\\ll m^{1/2}$?", "background": "Erdos and Sar\\'{a}nyi \\cite{ErSa59} proved that $f(m)\\gg m^{1/2}$.\nReferences\n\n\n[ErSa59] Erdos, P. and Sar\\'{a}nyi, Megjegyz\\'{e}sek egy versenyfeladathoz. Matematikai Lapok (1959).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2232, "problem_number": "EP-652", "title": "Erdős Problem #652", "statement": "Let $x_1,\\ldots,x_n\\in \\mathbb{R}^2$ and let $R(x_i)=\\#\\{ \\lvert x_j-x_i\\rvert : j\neq i\\}$, where the points are ordered such that $ R(x_1)\\leq \\cdots \\leq R(x_n). $ Let $\\alpha_k$ be minimal such that, for all large enough $n$, there exists a set of $n$ points with $R(x_k)<\\alpha_kn^{1/2}$. Is it true that $\\alpha_k\\to \\infty$ as $k\\to \\infty$?", "background": "It is trivial that $R(x_1)=1$ is possible, and that $R(x_2) \\ll n^{1/2}$ is also possible, but we always have $ R(x_1)R(x_2)\\gg n. $ Erdos originally conjectured that $R(x_3)/n^{1/2}\\to \\infty$ as $n\\to \\infty$, but Elekes proved that for every $k$ and $n$ sufficiently large there exists some set of $n$ points with $R(x_k)\\ll_k n^{1/2}$.\nMathialagan \\cite{Ma21} proved that given a set $P$ of $k$ points and a set $Q$ of $n$ points, with $2\\leq k\\leq n^{1/3}$, there exists a point in $P$ which determines $\\gg (kn)^{1/2}$ distances to points in $Q$. This immediately implies $R(x_k)\\gg (kn)^{1/2}$ for $2\\leq k\\leq n^{1/3}$.\nReferences\n\n\n[Ma21] Mathialagan, Surya, On bipartite distinct distances in the plane. Electron. J. Combin. (2021), Paper No. 4.33, 25.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2233, "problem_number": "EP-653", "title": "Erdős Problem #653", "statement": "Let $x_1,\\ldots,x_n\\in \\mathbb{R}^2$ and let $R(x_i)=\\#\\{ \\lvert x_j-x_i\\rvert : j\neq i\\}$, where the points are ordered such that $ R(x_1)\\leq \\cdots \\leq R(x_n). $ Let $g(n)$ be the maximum number of distinct values the $R(x_i)$ can take. Is it true that $g(n) \\geq (1-o(1))n$?", "background": "Erdos and Fishburn proved $g(n)>\\frac{3}{8}n$ and Csizmadia proved $g(n)>\\frac{7}{10}n$. Both groups proved $g(n) < n-cn^{2/3}$ for some constant $c>0$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2234, "problem_number": "EP-654", "title": "Erdős Problem #654", "statement": "Let $x_1,\\ldots,x_n\\in \\mathbb{R}^2$ with no four points on a circle. Must there exist some $x_i$ with at least $(1-o(1))n$ distinct distances to other $x_i$?", "background": "It is clear that every point has at least $\\frac{n-1}{3}$ distinct distances to other points in the set.\nIn \\cite{Er87b} and \\cite{ErPa90} Erdos and Pach ask this under the additional assumption that there are no three points on a line (so that the points are in general position), although they only ask the weaker question whether there is a lower bound of the shape $(\\tfrac{1}{3}+c)n$ for some constant $c>0$.\nThey suggest the lower bound $(1-o(1))n$ is true under the assumption that any circle around a point $x_i$ contains at most $2$ other $x_j$.\nReferences\n\n\n[Er87b] Erdos, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Si\\'{o}fok, 1985) (1987), 167-177.\n\n[ErPa90] Erdos, P. and Pach, J., Variations on the theme of repeated distances. Combinatorica (1990), 261--269.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2235, "problem_number": "EP-655", "title": "Erdős Problem #655", "statement": "Let $x_1,\\ldots,x_n\\in \\mathbb{R}^2$ be such that no circle whose centre is one of the $x_i$ contains three other points. Are there at least $ (1+c)\\frac{n}{2} $ distinct distances determined between the $x_i$, for some constant $c>0$ and all $n$ sufficiently large?", "background": "A problem of Erdos and Pach. It is easy to see that this assumption implies that there are at least $\\frac{n-1}{2}$ distinct distances determined by every point.\nZach Hunter has observed that taking $n$ points equally spaced on a circle disproves this conjecture. In the spirit of related conjectures of Erdos and others, presumably some kind of assumption that the points are in general position (e.g. no three on a line and no four on a circle) was intended.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2236, "problem_number": "EP-657", "title": "Erdős Problem #657", "statement": "Is it true that if $A\\subset \\mathbb{R}^2$ is a set of $n$ points such that every subset of $3$ points determines $3$ distinct distances (i.e. $A$ has no isosceles triangles) then $A$ must determine at least $f(n)n$ distinct distances, for some $f(n)\\to \\infty$?", "background": "In \\cite{Er73} Erdos attributes this problem (more generally in $\\mathbb{R}^k$) to himself and Davies. In \\cite{Er97e} he does not mention Davis, but says this problem was investigated by himself, F\"{u}redi, Ruzsa, and Pach.\nIn \\cite{Er73} Erdos says it is not even known in $\\mathbb{R}$ whether $f(n)\\to \\infty$. Sarosh Adenwalla has observed that this is equivalent to minimising the number of distinct differences in a set $A\\subset \\mathbb{R}$ of size $n$ without three-term arithmetic progressions. Dumitrescu \\cite{Du08} proved that, in these terms, $ (\\log n)^c \\leq f(n) \\leq 2^{O(\\sqrt{\\log n})} $ for some constant $c>0$.\nHunter observed in the comments that a result of Ruzsa coupled with standard tools of additive combinatorics (with details given by Alfaiz and Tang) allow recent progress on the size of subsets without three-term arithmetic progression (see \\cite{BlSi23} which improves slightly on the bounds due to Kelley and Meka \\cite{KeMe23}) yield $ 2^{c(\\log n)^{1/9}}\\leq f(n) $ for some constant $c>0$.\nStraus has observed that if $2^k\\geq n$ then there exist $n$ points in $\\mathbb{R}^k$ which contain no isosceles triangle and determine at most $n-1$ distances.\nSee also [135].\nReferences\n\n\n[BlSi23] T. F. Bloom and O. Sisask, An improvement to the Kelley-Meka bounds on three-term arithmetic progressions. arXiv:2309.02353 (2023).\n\n[Du08] Dumitrescu, Adrian, On distinct distances and {$\\lambda$}-free point sets. Discrete Math. (2008), 6533--6538.\n\n[Er73] Erdos, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138.\n\n[Er97e] Erdos, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537.\n\n[KeMe23] Kelley, Z. and Meka, R., Strong Bounds for 3-Progressions. arXiv:2302.05537 (2023).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2237, "problem_number": "EP-660", "title": "Erdős Problem #660", "statement": "Let $x_1,\\ldots,x_n\\in \\mathbb{R}^3$ be the vertices of a convex polyhedron. Are there at least $ (1-o(1))\\frac{n}{2} $ many distinct distances between the $x_i$?", "background": "For the similar problem in $\\mathbb{R}^2$ there are always at least $n/2$ distances, as proved by Altman \\cite{Al63} (see [93]). In \\cite{Er75f} Erdos claims that Altman proved that the vertices determine $\\gg n$ many distinct distances, but gives no reference.\nReferences\n\n\n[Al63] Altman, E., On a problem of P. Erdos. Amer. Math. Monthly (1963), 148-157.\n\n[Er75f] Erdos, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2238, "problem_number": "EP-661", "title": "Erdős Problem #661", "statement": "Are there, for all large $n$, some points $x_1,\\ldots,x_n,y_1,\\ldots,y_n\\in \\mathbb{R}^2$ such that the number of distinct distances $d(x_i,y_j)$ is $ o\\left(\\frac{n}{\\sqrt{\\log n}}\\right)? $ ", "background": "One can also ask this for points in $\\mathbb{R}^3$. In $\\mathbb{R}^4$ Lenz observed that there are $x_1,\\ldots,x_n,y_1,\\ldots,y_n\\in \\mathbb{R}^4$ such that $d(x_i,y_j)=1$ for all $i,j$, taking the points on two orthogonal circles.\nMore generally, if $F(2n)$ is the minimal number of such distances, and $f(2n)$ is minimal number of distinct distances between any $2n$ points in $\\mathbb{R}^2$, then is $F =o(f)$?\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2239, "problem_number": "EP-662", "title": "Erdős Problem #662", "statement": "Consider the triangular lattice with minimal distance between two points $1$. Denote by $f(t)$ the number of distances from any points $\\leq t$. For example $f(1)=6$, $f(\\sqrt{3})=12$, and $f(3)=18$.\nLet $x_1,\\ldots,x_n\\in \\mathbb{R}^2$ be such that $d(x_i,x_j)\\geq 1$ for all $i\neq j$. Is it true that, provided $n$ is sufficiently large depending on $t$, the number of distances $d(x_i,x_j)\\leq t$ is less than or equal to $f(t)$ with equality perhaps only for the triangular lattice?\nIn particular, is it true that the number of distances $\\leq \\sqrt{3}-\\epsilon$ is less than $1$?", "background": "A problem of Erdos, Lov\\'{a}sz, and Vesztergombi.\nThis is essentially verbatim the problem description in \\cite{Er97e}, but this does not make sense as written; there must be at least one typo. Suggestions about what this problem intends are welcome.\nErdos also goes on to write 'Perhaps the following stronger conjecture holds: Let $t_10$ and, for all large $n$, a pairwise balanced design such that $ \\lvert A_i\\rvert > n^{1/2}-C $ for all $1\\leq i\\leq m$?", "background": "A problem of Erdos and Larson \\cite{ErLa82}. In general, as Erdos asks in \\cite{Er97f}, find the slowest growing function $h$ such that, for all large $n$, there exists a pairwise balanced design with $ \\lvert A_i\\rvert > n^{1/2}-h(n) $ for all $1\\leq i\\leq m$.\nThe problem above asks whether $h(n)\\ll 1$. Erdos and Larson prove that $h(n) \\ll n^{1/2-c}$ for some constant $c>0$, and note this can be improved to $h(n)\\ll (\\log n)^2$ assuming Cramer-type bounds on the difference between consecutive primes.\nShrikhande and Singhi \\cite{ShSi85} have proved that the answer is no conditional on the conjecture that the order of every projective plane is a prime power (see [723]), by proving that every pairwise balanced design on $n$ points in which each block is of size $\\geq n^{1/2}-c$ can be embedded in a projective plane of order $n+i$ for some $i\\leq c+2$, if $n$ is sufficiently large.\nIn general, if $H(n)$ is the largest prime gap $\\leq n$, then the above reuslts show that, assuming the prime power conjecture, $H(n)\\asymp h(n)$.\nReferences\n\n\n[Er97f] Erdos, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10.\n\n[ErLa82] Erdos, P. and Larson, J., On pairwise balanced block designs with the sizes of blocks as uniform as possible. Annals of Discrete Mathematics (1982), 129-134.\n\n[ShSi85] S. S. Shrikhande and N. M. Singhi, On a problem of Erdos and Larson. Combinatorica (1985), 351-358.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2242, "problem_number": "EP-667", "title": "Erdős Problem #667", "statement": "Let $p,q\\geq 1$ be fixed integers. We define $H(n)=H(N;p,q)$ to be the largest $m$ such that any graph on $n$ vertices where every set of $p$ vertices spans at least $q$ edges must contain a complete graph on $m$ vertices.\nIs $ c(p,q)=\\liminf \\frac{\\log H(n)}{\\log n} $ a strictly increasing function of $q$ for $1\\leq q\\leq \\binom{p-1}{2}+1$?", "background": "A problem of Erdos, Faudree, Rousseau, and Schelp.\nWhen $q=1$ this corresponds exactly to the classical Ramsey problem, and hence for example $ \\frac{1}{p-1}\\leq c(p,1) \\leq \\frac{2}{p+1}. $ It is easy to see that if $q=\\binom{p-1}{2}+1$ then $c(p,q)=1$. Erdos, Faudree, Rousseau, and Schelp have shown that $c(p,\\binom{p-1}{2})\\leq 1/2$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2243, "problem_number": "EP-668", "title": "Erdős Problem #668", "statement": "Is it true that the number of incongruent sets of $n$ points in $\\mathbb{R}^2$ which maximise the number of unit distances tends to infinity as $n\\to\\infty$? Is it always $>1$ for $n>3$?", "background": "In fact this is $=1$ also for $n=4$, the unique example given by two equilateral triangles joined by an edge.\nComputational evidence of Engel, Hammond-Lee, Su, Varga, and Zs\\'{a}mboki \\cite{EHSVZ25} and Alexeev, Mixon, and Parshall \\cite{AMP25} suggests that this count is $=1$ for various other $5\\leq n\\leq 21$ (although these calculations were checking only up to graph isomorphism, rather than congruency).\nThe actual maximal number of unit distances is the subject of [90].\nReferences\n\n\n[AMP25] B. Alexeev, D. Mixon, and H. Parshall, The Erdos unit distance problem for small point sets. arXiv:2412.11914 (2025).\n\n[EHSVZ25] P. Engel, O. Hammond-Lee, Y. Su, D. Varga, and P. Zs\\'{a}mboki, Diverse beam search to find densest-known planar unit distance graphs. arXiv:2406.15317 (2025).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2244, "problem_number": "EP-669", "title": "Erdős Problem #669", "statement": "Let $F_k(n)$ be minimal such that for any $n$ points in $\\mathbb{R}^2$ there exist at most $F_k(n)$ many distinct lines passing through at least $k$ of the points, and $f_k(n)$ similarly but with lines passing through exactly $k$ points.\nEstimate $f_k(n)$ and $F_k(n)$ - in particular, determine $\\lim F_k(n)/n^2$ and $\\lim f_k(n)/n^2$.", "background": "Trivially $f_k(n)\\leq F_k(n)$ and $f_2(n)=F_2(n)=\\binom{n}{2}$. The problem with $k=3$ is the classical 'Orchard problem' of Sylvester. Burr, Gr\"{u}nbaum, and Sloane \\cite{BGS74} have proved that $ f_3(n)=\\frac{n^2}{6}-O(n) $ and $ F_3(n)=\\frac{n^2}{6}-O(n). $ There is a trivial upper bound of $F_k(n) \\leq \\binom{n}{2}/\\binom{k}{2}$, and hence $ \\lim F_k(n)/n^2 \\leq \\frac{1}{k(k-1)}. $ See also [101].\nReferences\n\n\n[BGS74] Burr, Stefan A. and Gr\"{u}nbaum, Branko and Sloane, N. J. A., The orchard problem. Geometriae Dedicata (1974), 397-424.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2245, "problem_number": "EP-670", "title": "Erdős Problem #670", "statement": "Let $A\\subseteq \\mathbb{R}^d$ be a set of $n$ points such that all pairwise distances differ by at least $1$. Is the diameter of $A$ at least $(1+o(1))n^2$?", "background": "The lower bound of $\\binom{n}{2}$ for the diameter is trivial. Erdos \\cite{Er97f} proved the claim when $d=1$.\nReferences\n\n\n[Er97f] Erdos, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2246, "problem_number": "EP-671", "title": "Erdős Problem #671", "statement": "Given $a_{i}^n\\in [-1,1]$ for all $1\\leq i\\leq n<\\infty$ we define $p_{i}^n$ as the unique polynomial of degree $n-1$ such that $p_{i}^n(a_{i}^n)=1$ and $p_{i}^n(a_{i'}^n)=0$ if $1\\leq i'\\leq n$ with $i\neq i'$. We similarly define $ \\mathcal{L}^nf(x) = \\sum_{1\\leq i\\leq n}f(a_i^n)p_i^n(x), $ the unique polynomial of degree $n-1$ which agrees with $f$ on $a_i^n$ for $1\\leq i\\leq n$ (that is, the sequence of Lagrange interpolation polynomials).", "background": "Is there such a sequence of $a_i^n$ such that for every continuous $f:[-1,1]\\to \\mathbb{R}$ there exists some $x\\in [-1,1]$ where $ \\limsup_{n\\to \\infty} \\sum_{1\\leq i\\leq n}\\lvert p_{i}^n(x)\\rvert=\\infty $ and yet $ \\mathcal{L}^nf(x) \\to f(x)? $ Is there such a sequence such that $ \\limsup_{n\\to \\infty} \\sum_{1\\leq i\\leq n}\\lvert p_{i}^n(x)\\rvert=\\infty $ for every $x\\in [-1,1]$ and yet for every continuous $f:[-1,1]\\to \\mathbb{R}$ there exists $x\\in [-1,1]$ with $ \\mathcal{L}^nf(x) \\to f(x)? $ \nBernstein \\cite{Be31} proved that for any choice of $a_i^n$ there exists $x_0\\in [-1,1]$ such that $ \\limsup_{n\\to \\infty} \\sum_{1\\leq i\\leq n}\\lvert p_{i}^n(x)\\rvert=\\infty. $ Erdos and V\\'{e}rtesi \\cite{ErVe80} proved that for any choice of $a_i^n$ there exists a continuous $f:[-1,1]\\to \\mathbb{R}$ such that $ \\limsup_{n\\to \\infty} \\lvert \\mathcal{L}^nf(x)\\rvert=\\infty $ for almost all $x\\in [-1,1]$.\nReferences\n\n\n[Be31] S. Bernstein, Sur la limitation des valeurs d'un polynome $P_n(x)$ de degr\\'{e} n sur tout un segment par ses valeurs en $(n+1)$ points du segment. Izv. Akad. Nauk. SSSR (1931), 1025-1050.\n\n[ErVe80] Erdos, P. and V\\'{e}rtesi, P., On the almost everywhere divergence of Lagrange\ninterpolatory polynomials for arbitrary system of nodes. Acta Math. Acad. Sci. Hungar. (1980), 71-89.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2247, "problem_number": "EP-675", "title": "Erdős Problem #675", "statement": "We say that $A\\subset \\mathbb{N}$ has the translation property if, for every $n$, there exists some integer $t_n\\geq 1$ such that, for all $1\\leq a\\leq n$, $ a\\in A\\quad\\textrm{ if and only if }\\quad a+t_n\\in A. $ {UL}", "background": "{LI}Does the set of the sums of two squares have the translation property?{/LI}\n{LI}If we partition all primes into $P\\sqcup Q$, such that each set contains $\\gg x/\\log x$ many primes $\\leq x$ for all large $x$, then can the set of integers only divisible by primes from $P$ have the translation property?{/LI}\n{LI}If $A$ is the set of squarefree numbers then how fast does the minimal such $t_n$ grow? Is it true that $t_n>\\exp(n^c)$ for some constant $c>0$?{/LI}\n{/UL}\nElementary sieve theory implies that the set of squarefree numbers has the translation property.\nMore generally, Brun's sieve can be used to prove that if $B\\subseteq \\mathbb{N}$ is a set of pairwise coprime integers with $\\sum_{b0$. Erdos \\cite{Er79} believed it is 'rather unlikely' that all large integers are of this form.\nWhat if the condition that $p$ is prime is omitted? Selfridge and Wagstaff made a 'preliminary computer search' and suggested that there are infinitely many $n$ not of this form even without the condition that $p$ is prime. It should be true that the number of exceptions in $[1,x]$ is $1$? Erdos expects very few (and none when $l\\geq k$).\nThe only solutions Erdos knew were $M(4,3)=M(13,2)$ and $M(3,4)=M(19,2)$.\nIn \\cite{Er79d} Erdos conjectures the stronger fact that (aside from a finite number of exceptions) if $k>2$ and $m\\geq n+k$ then $\\prod_{i\\leq k}(n+i)$ and $\\prod_{i\\leq k}(m+i)$ cannot have the same set of prime factors.\nSee also [678], [686], and [850].\nThis is discussed in problem B35 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Er79d] Erdos, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2250, "problem_number": "EP-679", "title": "Erdős Problem #679", "statement": "Let $\\epsilon>0$ and $\\omega(n)$ count the number of distinct prime factors of $n$. Are there infinitely many values of $n$ such that $ \\omega(n-k) < (1+\\epsilon)\\frac{\\log k}{\\log\\log k} $ for all $k0$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2251, "problem_number": "EP-680", "title": "Erdős Problem #680", "statement": "Is it true that, for all sufficiently large $n$, there exists some $k$ such that $ p(n+k)>k^2+1, $ where $p(m)$ denotes the least prime factor of $m$?\nCan one prove this is false if we replace $k^2+1$ by $e^{(1+\\epsilon)\\sqrt{k}}+C_\\epsilon$, for all $\\epsilon>0$, where $C_\\epsilon>0$ is some constant?", "background": "This follows from 'plausible assumptions on the distribution of primes' (as does the question with $k^2$ replaced by $k^d$ for any $d$); the challenge is to prove this unconditionally.\nErdos observed that Cramer's conjecture $ \\limsup_{k\\to \\infty} \\frac{p_{k+1}-p_k}{(\\log k)^2}=1 $ implies that for all $\\epsilon>0$ and all sufficiently large $n$ there exists some $k$ such that $ p(n+k)>e^{(1-\\epsilon)\\sqrt{k}}. $ There is now evidence, however, that Cramer's conjecture is false; a more refined heuristic by Granville \\cite{Gr95} suggests this $\\limsup$ is $2e^{-\\gamma}\\approx 1.119\\cdots$, and so perhaps the $1+\\epsilon$ in the second question should be replaced by $2e^{-\\gamma}+\\epsilon$.\nSee also [681] and [682].\nReferences\n\n\n[Gr95] Granville, Andrew, Harald {C}ram\\'{e}r and the distribution of prime numbers. Scand. Actuar. J. (1995), 12--28.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2252, "problem_number": "EP-681", "title": "Erdős Problem #681", "statement": "Is it true that for all large $n$ there exists $k$ such that $n+k$ is composite and $ p(n+k)>k^2, $ where $p(m)$ is the least prime factor of $m$?", "background": "Related to questions of Erdos, Eggleton, and Selfridge. This may be true with $k^2$ replaced by $k^d$ for any $d$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2253, "problem_number": "EP-683", "title": "Erdős Problem #683", "statement": "Is it true that for every $1\\leq k\\leq n$ the largest prime divisor of $\\binom{n}{k}$, say $P(\\binom{n}{k})$, satisfies $ P\\left(\\binom{n}{k}\\right)\\geq \\min(n-k+1, k^{1+c}) $ for some constant $c>0$?", "background": "A theorem of Sylvester and Schur (see \\cite{Er34}) states that $P(\\binom{n}{k})>k$ if $k\\leq n/2$. Erdos \\cite{Er55d} proved that there exists some $c>0$ such that, whenever $k\\leq n/2$, $ P\\left(\\binom{n}{k}\\right)\\gg k\\log k. $ Erdos \\cite{Er79d} writes it 'seems certain' that this holds for every $c>0$, with only a finite number of exceptions (depending on $c$). Standard heuristics on prime gaps suggest that the largest prime divisor of $\\binom{n}{k}$ is, for $k\\leq n/2$, in fact $ >e^{c\\sqrt{k}} $ for some constant $c>0$.\nThis is essentially equivalent to [961].\nReferences\n\n\n[Er34] Erdos, Paul, A {T}heorem of {S}ylvester and {S}chur. J. London Math. Soc. (1934), 282--288.\n\n[Er55d] Erdos, P., On consecutive integers. Nieuw Arch. Wisk. (3) (1955), 124--128.\n\n[Er79d] Erdos, P., Some unconventional problems in number theory. Acta Math. Acad. Sci. Hungar. (1979), 71-80.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2254, "problem_number": "EP-684", "title": "Erdős Problem #684", "statement": "For $0\\leq k\\leq n$ write $ \\binom{n}{k} = uv $ where the only primes dividing $u$ are in $[2,k]$ and the only primes dividing $v$ are in $(k,n]$.\nLet $f(n)$ be the smallest $k$ such that $u>n^2$. Give bounds for $f(n)$.", "background": "A classical theorem of Mahler states that for any $\\epsilon>0$ and integers $k$ and $l$ then, writing $ (n+1)\\cdots (n+k) = ab $ where the only primes dividing $a$ are $\\leq l$ and the only primes dividing $b$ are $>l$, we have $a < n^{1+\\epsilon}$ for all sufficiently large (depending on $\\epsilon,k,l$) $n$.\nMahler's theorem implies $f(n)\\to \\infty$ as $n\\to \\infty$, but is ineffective, and so gives no bounds on the growth of $f(n)$.\nOne can similarly ask for estimates on the smallest integer $f(n,k)$ such that if $m$ is the factor of $\\binom{n}{k}$ containing all primes $\\leq f(n,k)$ then $m > n^2$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2255, "problem_number": "EP-685", "title": "Erdős Problem #685", "statement": "Let $\\epsilon>0$ and $n$ be large depending on $\\epsilon$. Is it true that for all $n^\\epsilon\\frac{\\log \\binom{n}{k}}{\\log n}, $ and this inequality becomes (asymptotic) equality if $k>n^{1-o(1)}$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2256, "problem_number": "EP-686", "title": "Erdős Problem #686", "statement": "Can every integer $N\\geq 2$ be written as $ N=\\frac{\\prod_{1\\leq i\\leq k}(m+i)}{\\prod_{1\\leq i\\leq k}(n+i)} $ for some $k\\geq 2$ and $m\\geq n+k$?", "background": "If $n$ and $k$ are fixed then can one say anything about the set of integers so represented?\nSee also [677].\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2257, "problem_number": "EP-687", "title": "Erdős Problem #687", "statement": "Let $Y(x)$ be the maximal $y$ such that there exists a choice of congruence classes $a_p$ for all primes $p\\leq x$ such that every integer in $[1,y]$ is congruent to at least one of the $a_p\\pmod{p}$.\nGive good estimates for $Y(x)$. In particular, can one prove that $Y(x)=o(x^2)$ or even $Y(x)\\ll x^{1+o(1)}$?", "background": "This function (associated with Jacobsthal) is closely related to the problem of gaps between primes (see [4]). The best known upper bound is due to Iwaniec \\cite{Iw78}, $ Y(x) \\ll x^2. $ The best lower bound is due to Ford, Green, Konyagin, Maynard, and Tao \\cite{FGKMT18}, $ Y(x) \\gg x\\frac{\\log x\\log\\log\\log x}{\\log\\log x}, $ improving on a previous bound of Rankin \\cite{Ra38}.\nMaier and Pomerance have conjectured that $Y(x)\\ll x(\\log x)^{2+o(1)}$.\nIn \\cite{Er80} he writes 'It is not clear who first formulated this problem - probably many of us did it independently. I offer the maximum of \\$1000 dollars and $1/2$ my total savings for clearing up of this problem.'\nIn \\cite{Er80} Erdos also asks about a weaker variant in which all except $o(y/\\log y)$ of the integers in $[1,y]$ are congruent to at least one of the $a_p\\pmod{p}$, and in particular asks if the answer is very different.\nSee also [688] and [689]. A more general Jacobsthal function is the focus of [970].\nReferences\n\n\n[Er80] Erdos, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.\n\n[FGKMT18] Ford, Kevin and Green, Ben and Konyagin, Sergei and Maynard, James and Tao, Terence, Long gaps between primes. J. Amer. Math. Soc. (2018), 65-105.\n\n[Iw78] Iwaniec, Henryk, On the problem of {J}acobsthal. Demonstratio Math. (1978), 225--231.\n\n[Ra38] Rankin, R. A., The Difference between Consecutive Prime Numbers. J. London Math. Soc. (1938), 242-247.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2258, "problem_number": "EP-688", "title": "Erdős Problem #688", "statement": "Define $\\epsilon_n$ to be maximal such that there exists some choice of congruence class $a_p$ for all primes $n^{\\epsilon_n}\\alpha$.\nTenenbaum notes in \\cite{Te96} that this is certainly not true as written since if the $n_j$ grow sufficiently quickly then this sequence is never Behrend, for any choice of $\\eta_k$. He then writes 'we understand from subsequent discussions with Erdos that he had actually in mind a two-sided condition on' $n_{j+1}/n_j$.\nTenenbaum \\cite{Te96} proves this conjecture: if there are constants $1\\log 2$.\nReferences\n\n\n[Te96] Tenenbaum, G., On block {B}ehrend sequences. Math. Proc. Cambridge Philos. Soc. (1996), 355--367.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2262, "problem_number": "EP-693", "title": "Erdős Problem #693", "statement": "Let $k\\geq 2$ and $n$ be sufficiently large depending on $k$. Let $A=\\{a_1n^{1/2}$?{/LI}\n{LI} Is it true that, for every composite $n$, $ f(n) \\ll_A \\frac{n}{(\\log n)^A} $ for every $A>0$?{/LI}\n{/UL}", "background": "A problem of Erdos and Szekeres. It is easy to see that $f(n)\\leq n/P(n)$ for composite $n$, since if $j=p^k$ where $p^k\\mid n$ and $p^{k+1}\nmid n$ then $\\textrm{gcd}\\left(n,\\binom{n}{j}\\right)=n/p^k$. This implies $ f(n) \\leq (1+o(1))\\frac{n}{\\log n}. $ It is known that $f(n)=n/P(n)$ when $n$ is the product of two primes. Another example is $n=30$.\nFor the second problem, it is easy to see that for any $n$ we have $f(n)\\geq p(n)$, where $p(n)$ is the smallest prime dividing $n$, and hence there are infinitely many $n$ (those $=p^2)$ such that $f(n)\\geq n^{1/2}$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2267, "problem_number": "EP-701", "title": "Erdős Problem #701", "statement": "Let $\\mathcal{F}$ be a family of sets closed under taking subsets (i.e. if $B\\subseteq A\\in\\mathcal{F}$ then $B\\in \\mathcal{F}$). There exists some element $x$ such that whenever $\\mathcal{F}'\\subseteq \\mathcal{F}$ is an intersecting subfamily we have $ \\lvert \\mathcal{F}'\\rvert \\leq \\lvert \\{ A\\in \\mathcal{F} : x\\in A\\}\\rvert. $ ", "background": "A problem of Chv\\'{a}tal \\cite{Ch74}, who proved it replacing the closed under subsets condition with the (stronger) condition that, assuming all sets in $\\mathcal{F}$ are subsets of $\\{1,\\ldots,n\\}$, whenever $A\\in \\mathcal{F}$ and there is an injection $f:B\\to A$ such that $x\\leq f(x)$ for all $x\\in B$, then $B\\in \\mathcal{F}$.\nSterboul \\cite{St74} proved this when, letting $\\mathcal{G}$ be the maximal sets (under inclusion) in $\\mathcal{F}$, all sets in $\\mathcal{G}$ have the same size, $\\lvert A\\cap B\\rvert\\leq 1$ for all $A\neq B\\in \\mathcal{G}$, and at least two sets in $\\mathcal{G}$ have non-empty intersection.\nFrankl and Kupavskii \\cite{FrKu23} have proved this when $\\mathcal{F}$ has covering number $2$.\nBorg \\cite{Bo11} has proposed a weighted generalisation of this conjecture, which he proves under certain additional assumptions.\nReferences\n\n\n[Bo11] Borg, Peter, On Chv\\'{a}tal's conjecture and a conjecture on families of\nsigned sets. European J. Combin. (2011), 140-145.\n\n[Ch74] Chv\\'{a}tal, V., Intersecting families of edges in hypergraphs having the\nhereditary property. (1974), 61-66.\n\n[FrKu23] Frankl, Peter and Kupavskii, Andrey, Perfect matchings in down-sets. Discrete Math. (2023), Paper No. 113323, 7.\n\n[St74] Sterboul, F., Sur une conjecture de V. Chv\\'{a}tal. (1974), 152-164.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2268, "problem_number": "EP-704", "title": "Erdős Problem #704", "statement": "Let $G_n$ be the unit distance graph in $\\mathbb{R}^n$, with two vertices joined by an edge if and only if the distance between them is $1$.\nEstimate the chromatic number $\\chi(G_n)$. Does it grow exponentially in $n$? Does $ \\lim_{n\\to \\infty}\\chi(G_n)^{1/n} $ exist?", "background": "A generalisation of the Hadwiger-Nelson problem (which addresses $n=2$). Frankl and Wilson \\cite{FrWi81} proved exponential growth: $ \\chi(G_n) \\geq (1+o(1))1.2^n. $ The trivial colouring (by tiling with cubes) gives $ \\chi(G_n) \\leq (2+\\sqrt{n})^n. $ Larman and Rogers \\cite{LaRo72} improved this to $ \\chi(G_n) \\leq (3+o(1))^n, $ and conjecture the truth may be $(2^{3/2}+o(1))^n$. Prosanov \\cite{Pr20} has given an alternative proof of this upper bound.\nSee also [508], [705], and [706].\nReferences\n\n\n[FrWi81] Frankl, P. and Wilson, R. M., Intersection theorems with geometric consequences. Combinatorica (1981), 357-368.\n\n[LaRo72] Larman, D. G. and Rogers, C. A., The realization of distances within sets in Euclidean space. Mathematika (1972), 1-24.\n\n[Pr20] Prosanov, Roman, A new proof of the Larman-Rogers upper bound for the\nchromatic number of the Euclidean space. Discrete Appl. Math. (2020), 115-120.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2269, "problem_number": "EP-705", "title": "Erdős Problem #705", "statement": "Let $G$ be a finite unit distance graph in $\\mathbb{R}^2$ (i.e. the vertices are a finite collection of points in $\\mathbb{R}^2$ and there is an edge between two points if and only if the distance between them is $1$).\nIs there some $k$ such that if $G$ has girth $\\geq k$ (i.e. $G$ contains no cycles of length $p_\\ell^2$.\nGallai was the first to consider problems of this type, and observed that $g(2)=2$ and $g(3)\\geq 4$.\nIn \\cite{Er92c} Erdos offers '100 dollars or 1000 rupees', whichever is more, for a proof or disproof. (In 1992 1000 rupees was worth approximately \\$38.60.)\nErdos and Sur\\'{a}nyi similarly asked what is the smallest $c_n\\geq 1$ such that in any interval $I\\subset [0,\\infty)$ of length $c_n\\max(A)$ there exists some $B\\subseteq I\\cap \\mathbb{N}$ with $\\lvert B\\rvert=n$ such that $ \\prod_{a\\in A} a \\mid \\prod_{b\\in B}b. $ They prove $c_2=1$ and $c_3=\\sqrt{2}$, but have no good upper or lower bounds in general.\nSee also [709].\nReferences\n\n\n[Er92c] Erd\"{o}s, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. (1992), 34-50.\n\n[ErSu59] Erdos, P\\'{a}l and Sur\\'{a}nyi, J\\'{a}nos, Bemerkungen zu einer Aufgabe eines mathematischen\n{W}ettbewerbs. Mat. Lapok (1959), 39-48.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2272, "problem_number": "EP-709", "title": "Erdős Problem #709", "statement": "Let $f(n)$ be minimal such that, for any $A=\\{a_1,\\ldots,a_n\\}\\subseteq [2,\\infty)\\cap\\mathbb{N}$ of size $n$, in any interval $I$ of $f(n)\\max(A)$ consecutive integers there exist distinct $x_1,\\ldots,x_n\\in I$ such that $a_i\\mid x_i$.\nObtain good bounds for $f(n)$, or even an asymptotic formula.", "background": "A problem of Erdos and Sur\\'{a}nyi \\cite{ErSu59}, who proved $ (\\log n)^c \\ll f(n) \\ll n^{1/2} $ for some constant $c>0$.\nSee also [708].\nReferences\n\n\n[ErSu59] Erdos, P\\'{a}l and Sur\\'{a}nyi, J\\'{a}nos, Bemerkungen zu einer Aufgabe eines mathematischen\n{W}ettbewerbs. Mat. Lapok (1959), 39-48.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2273, "problem_number": "EP-710", "title": "Erdős Problem #710", "statement": "Let $f(n)$ be minimal such that in $(n,n+f(n))$ there exist distinct integers $a_1,\\ldots,a_n$ such that $k\\mid a_k$ for all $1\\leq k\\leq n$. Obtain an asymptotic formula for $f(n)$.", "background": "A problem of Erdos and Pomerance \\cite{ErPo80}, who proved $ (2/\\sqrt{e}+o(1))n\\left(\\frac{\\log n}{\\log\\log n}\\right)^{1/2}\\leq f(n)\\leq (1.7398\\cdots+o(1))n(\\log n)^{1/2}. $ In \\cite{Er92c} Erdos offered 2000 rupees for an asymptotic formula; for uniform comparison across prizes I have converted this using the 1992 exchange rates.\nSee also [711].\nReferences\n\n\n[Er92c] Erd\"{o}s, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. (1992), 34-50.\n\n[ErPo80] P. Erdos and C. Pomerance, Matching the natural numbers up to $n$ with distinct multiples of another interval. Indigationes Math. (1980), 147-151.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2274, "problem_number": "EP-711", "title": "Erdős Problem #711", "statement": "Let $f(n,m)$ be minimal such that in $(m,m+f(n,m))$ there exist distinct integers $a_1,\\ldots,a_n$ such that $k\\mid a_k$ for all $1\\leq k\\leq n$. Prove that $ \\max_m f(n,m) \\leq n^{1+o(1)} $ and that $ \\max_m (f(n,m)-f(n,n))\\to \\infty. $ ", "background": "A problem of Erdos and Pomerance \\cite{ErPo80}, who proved that $ \\max_m f(n,m) \\ll n^{3/2} $ and $ n\\left(\\frac{\\log n}{\\log\\log n}\\right)^{1/2} \\ll f(n,n)\\ll n(\\log n)^{1/2}. $ In \\cite{Er92c} Erdos offered 1000 rupees for a proof of either; for uniform comparison across prizes I have converted this using the 1992 exchange rates.\nvan Doorn \\cite{vD26} has provided an affirmative answer to the second question, proving that, for all large $n$, there exists $m=m(n)$ such that $ f(n,m)-f(n,n) \\gg \\frac{\\log n}{\\log\\log n}n. $ See also [710].\nReferences\n\n\n[Er92c] Erd\"{o}s, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. (1992), 34-50.\n\n[ErPo80] P. Erdos and C. Pomerance, Matching the natural numbers up to $n$ with distinct multiples of another interval. Indigationes Math. (1980), 147-151.\n\n[vD26] W. van Doorn, On the length of an interval that contains distinct multiples of the first $n$ positive integers. Integers (2026), #A7.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2275, "problem_number": "EP-712", "title": "Erdős Problem #712", "statement": "Determine, for any $k>r>2$, the value of $ \\frac{\\mathrm{ex}_r(n,K_k^r)}{\\binom{n}{r}}, $ where $\\mathrm{ex}_r(n,K_k^r)$ is the largest number of $r$-edges which can placed on $n$ vertices so that there exists no set of $k$ vertices which is covered by all $\\binom{k}{r}$ possible $r$-edges.", "background": "Tur\\'{an proved} that, when $r=2$, this limit is $ \\frac{1}{2}\\left(1-\\frac{1}{k-1}\\right). $ Erdos \\cite{Er81} offered \\$500 for the determination of this value for any fixed $k>r>2$, and \\$1000 for 'clearing up the whole set of problems'.\nSee also [500] for the case $r=3$ and $k=4$.\nReferences\n\n\n[Er81] Erdos, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2276, "problem_number": "EP-713", "title": "Erdős Problem #713", "statement": "Is it true that, for every bipartite graph $G$, there exists some $\\alpha\\in [1,2)$ and $c>0$ such that $ \\mathrm{ex}(n;G)\\sim cn^\\alpha? $ Must $\\alpha$ be rational?", "background": "A problem of Erdos and Simonovits. Erdos sometimes asked this in the weaker version with just $ \\mathrm{ex}(n;G)\\asymp n^{\\alpha}. $ Erdos \\cite{Er67d} had initially conjectured that, for any bipartite graph $G$, $\\mathrm{ex}(n;G)\\sim cn^{\\alpha}$ for some constant $c>0$ and $\\alpha$ of the shape $1+\\frac{1}{k}$ or $2-\\frac{1}{k}$ for some integer $k\\geq 2$. This was disproved by Erdos and Simonovits \\cite{ErSi70}.\nThe analogous statement is not true for hypergraphs, as shown by Frankl and F\"{u}redi \\cite{FrFu87}, who proved that if $G$ is the $5$-uniform hypergraph on $8$ vertices with edges $\\{12346,12457,12358\\}$ then $\\mathrm{ex}(n;G)=o(n^5)$ but $\\mathrm{ex}(n;G)\neq O(n^c)$ for any $c<5$.\nA simplified proof was given by F\"{u}redi and Gerbner \\cite{FuGe21}, who extended it to a counterexample for all $k\\geq 5$. It remains open whether it is true for $k=3$ and $k=4$ (though F\"{u}redi and Gerbner conjecture it is not).\nSee also [571].\nReferences\n\n\n[Er67d] Erdos, P., Some recent results on extremal problems in graph theory.\n{R}esults. (1967), 117--123 (English); pp. 124--130 (French).\n\n[ErSi70] Erdos, P. and Simonovits, M., Some extremal problems in graph theory. Combinatorial theory and its applications, I-III (Proc. Colloq., Balatonf\"{u}red, 1969) (1970), 377-390.\n\n[FrFu87] Frankl, P. and F\"uredi, Z., Exact solution of some {T}ur\\'an-type problems. J. Combin. Theory Ser. A (1987), 226--262.\n\n[FuGe21] F\"uredi, Zolt\\'an and Gerbner, D\\'aniel, Hypergraphs without exponents. J. Combin. Theory Ser. A (2021), Paper No. 105517, 9.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2277, "problem_number": "EP-714", "title": "Erdős Problem #714", "statement": "Is it true that $ \\mathrm{ex}(n; K_{r,r}) \\gg n^{2-1/r}? $ ", "background": "K\"{o}v\\'{a}ri, S\\'{o}s, and Tur\\'{a}n \\cite{KST54} proved $ \\mathrm{ex}(n; K_{r,r}) \\ll n^{2-1/r} $ for all $r\\geq 2$. Brown \\cite{Br66} and, independently, Erdos, R\\'{e}nyi, and S\\'{o}s \\cite{ERS66}, proved the conjectured lower bound when $r=3$.\nWhen $r=2$ it is known that $ \\mathrm{ex}(n;K_{2,2})=\\left(\\frac{1}{2}+o(1)\\right)n^{3/2} $ (see [768], since $K_{2,2}=C_4$).\nSee also [147].\nReferences\n\n\n[Br66] Brown, W. G., On graphs that do not contain a Thomsen graph. Canad. Math. Bull. (1966), 281-285.\n\n[ERS66] Erdos, P. and R\\'{e}nyi, A. and S\\'os, V. T., On a problem of graph theory. Studia Sci. Math. Hungar. (1966), 215--235.\n\n[KST54] K\"{o}vari, T. and S\\'{o}s, V. T. and Tur\\'{a}n, P., On a problem of K. Zarankiewicz. Colloq. Math. (1954), 50-57.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2278, "problem_number": "EP-719", "title": "Erdős Problem #719", "statement": "Let $\\mathrm{ex}_r(n;K_{r+1}^r)$ be the maximum number of $r$-edges that can be placed on $n$ vertices without forming a $K_{r+1}^r$ (the $r$-uniform complete graph on $r+1$ vertices).\nIs every $r$-hypergraph $G$ on $n$ vertices the union of at most $\\mathrm{ex}_{r}(n;K_{r+1}^r)$ many copies of $K_r^r$ and $K_{r+1}^r$, no two of which share a $K_r^r$?\",\n \"difficulty\": \"L1\"\n},{", "background": "", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2279, "problem_number": "EP-724", "title": "Erdős Problem #724", "statement": "Let $f(n)$ be the maximum number of mutually orthogonal Latin squares of order $n$. Is it true that $ f(n) \\gg n^{1/2}? $ ", "background": "Euler conjectured that $f(n)=1$ when $n\\equiv 2\\pmod{4}$, but this was disproved by Bose, Parker, and Shrikhande \\cite{BPS60} who proved $f(n)\\geq 2$ for $n\\geq 7$.\nChowla, Erdos, and Straus \\cite{CES60} proved $f(n) \\gg n^{1/91}$. Wilson \\cite{Wi74} proved $f(n) \\gg n^{1/17}$. Beth \\cite{Be83c} proved $f(n) \\gg n^{1/14.8}$.\nThe sequence of $f(n)$ is A001438 in the OEIS.\nReferences\n\n\n[BPS60] Bose, R. C. and Shrikhande, S. S. and Parker, E. T., Further results on the construction of mutually orthogonal\nLatin squares and the falsity of Euler's conjecture. Canadian J. Math. (1960), 189-203.\n\n[Be83c] Beth, Thomas, Eine Bemerkung zur Absch\"{a}tzung der Anzahl orthogonaler\nlateinischer Quadrate mittels Siebverfahren. Abh. Math. Sem. Univ. Hamburg (1983), 284-288.\n\n[CES60] Chowla, S. and Erdos, P. and Straus, E. G., On the maximal number of pairwise orthogonal Latin squares\nof a given order. Canadian J. Math. (1960), 204-208.\n\n[Wi74] Wilson, Richard M., Concerning the number of mutually orthogonal Latin squares. Discrete Math. (1974), 181-198.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2280, "problem_number": "EP-725", "title": "Erdős Problem #725", "statement": "Give an asymptotic formula for the number of $k\\times n$ Latin rectangles.", "background": "Erdos and Kaplansky \\cite{ErKa46} proved the count is $ \\sim e^{-\\binom{k}{2}}(n!)^k $ when $k=o((\\log n)^{3/2-\\epsilon})$. Yamamoto \\cite{Ya51} extended this to $k\\leq n^{1/3-o(1)}$.\nThe count of such Latin rectangles is A001009 in the OEIS.\nReferences\n\n\n[ErKa46] Erd\"{o}s, Paul and Kaplansky, Irving, The asymptotic number of Latin rectangles. Amer. J. Math. (1946), 230-236.\n\n[Ya51] Yamamoto, Koichi, On the asymptotic number of Latin rectangles. Jpn. J. Math. (1951), 113-119.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2281, "problem_number": "EP-726", "title": "Erdős Problem #726", "statement": "As $n\\to \\infty$ ranges over integers $ \\sum_{p\\leq n}1_{n\\in (p/2,p)\\pmod{p}}\\frac{1}{p}\\sim \\frac{\\log\\log n}{2}. $ ", "background": "A conjecture of Erdos, Graham, Ruzsa, and Straus \\cite{EGRS75}. For comparison the classical estimate of Mertens states that $ \\sum_{p\\leq n}\\frac{1}{p}\\sim \\log\\log n. $ By $n\\in (p/2,p)\\pmod{p}$ we mean $n\\equiv r\\pmod{p}$ for some integer $r$ with $p/20$).\nErdos \\cite{Er68c} proved that if $a!b!\\mid n!$ then $a+b\\leq n+O(\\log n)$.\nReferences\n\n\n[Ba29] H. Balakran, On the values of $n$ which make $(2n)!/(n+1)!(n+1)!$ an integer. J. Indian Math. Soc. (1929), 97-100.\n\n[EGRS75] Erdos, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., On the prime factors of $(\\sp{2n}\\sb{n})$. Math. Comp. (1975), 83-92.\n\n[Er68c] P. Erdos, Aufgabe 557. Elemente Math. (1968), 111-113.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2283, "problem_number": "EP-730", "title": "Erdős Problem #730", "statement": "Are there infinitely many pairs of integers $n\neq m$ such that $\\binom{2n}{n}$ and $\\binom{2m}{m}$ have the same set of prime divisors?", "background": "A problem of Erdos, Graham, Ruzsa, and Straus \\cite{EGRS75}, who believed there is 'no doubt' that the answer is yes.\nFor example $(87,88)$ and $(607,608)$. Those $n$ such that there exists some suitable $m>n$ are listed as A129515 in the OEIS.\nA triple of such $n$ for which $\\binom{2n}{n}$ all share the same set of prime divisors is $(10003,10004,10005)$. It is not known whether there are such pairs of the shape $(n,n+k)$ for every $k\\geq 1$.\nReferences\n\n\n[EGRS75] Erdos, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., On the prime factors of $(\\sp{2n}\\sb{n})$. Math. Comp. (1975), 83-92.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2284, "problem_number": "EP-731", "title": "Erdős Problem #731", "statement": "Find some reasonable function $f(n)$ such that, for almost all integers $n$, the least integer $m$ such that $m\nmid \\binom{2n}{n}$ satisfies $ m\\sim f(n). $ ", "background": "A problem of Erdos, Graham, Ruzsa, and Straus \\cite{EGRS75}, who say it is 'not hard to show that', for almost all $n$, the minimal such $m$ satisfies $ m=\\exp((\\log n)^{1/2+o(1)}). $ \nReferences\n\n\n[EGRS75] Erdos, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., On the prime factors of $(\\sp{2n}\\sb{n})$. Math. Comp. (1975), 83-92.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2285, "problem_number": "EP-734", "title": "Erdős Problem #734", "statement": "Find, for all large $n$, a non-trivial pairwise balanced block design $A_1,\\ldots,A_m\\subseteq \\{1,\\ldots,n\\}$ such that, for all $t$, there are $O(n^{1/2})$ many $i$ such that $\\lvert A_i\\rvert=t$.", "background": "$A_1,\\ldots,A_m$ is a pairwise balanced block design if every pair in $\\{1,\\ldots,n\\}$ is contained in exactly one of the $A_i$.\nErdos \\cite{Er81} writes 'this will be probably not be very difficult to prove but so far I was not successful'.\nErdos and de Bruijn \\cite{dBEr48} proved that if $A_1,\\ldots,A_m\\subseteq \\{1,\\ldots,n\\}$ is a pairwise balanced block design then $m\\geq n$, and this implies there must be some $t$ such that there are $\\gg n^{1/2}$ many $t$ with $\\lvert A_i\\rvert=t$.\nReferences\n\n\n[Er81] Erdos, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.\n\n[dBEr48] de Bruijn, N. G. and Erdos, P., On a combinatorial problem. Nederl. Akad. Wetensch., Proc. (1948), 1277--1279 = Indagationes Math. 10, 421--423.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2286, "problem_number": "EP-738", "title": "Erdős Problem #738", "statement": "If $G$ has infinite chromatic number and is triangle-free (contains no $K_3$) then must $G$ contain every tree as an induced subgraph?\n\",\n \"difficulty\": \"L1\"\n},{", "background": "", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2287, "problem_number": "EP-740", "title": "Erdős Problem #740", "statement": "Let $\\mathfrak{m}$ be an infinite cardinal and $G$ be a graph with chromatic number $\\mathfrak{m}$. Let $r\\geq 1$. Must $G$ contain a subgraph of chromatic number $\\mathfrak{m}$ which does not contain any odd cycle of length $\\leq r$?", "background": "A question of Erdos and Hajnal. R\"{o}dl proved this is true if $\\mathfrak{m}=\\aleph_0$ and $r=3$ (see [108] for the finitary version).\nMore generally, Erdos and Hajnal asked must there exist (for every cardinal $\\mathfrak{m}$ and integer $r$) some $f_r(\\mathfrak{m})$ such that every graph with chromatic number $\\geq f_r(\\mathfrak{m})$ contains a subgraph with chromatic number $\\mathfrak{m}$ with no odd cycle of length $\\leq r$?\nErdos \\cite{Er95d} claimed that even the $r=3$ case of this is open: must every graph with sufficiently large chromatic number contain a triangle free graph with chromatic number $\\mathfrak{m}$?\nIn \\cite{Er81} Erdos also asks the same question but with girth (i.e. the subgraph does not contain any cycle at all of length $\\leq C$).\nReferences\n\n\n[Er81] Erdos, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.\n\n[Er95d] Erdos, Paul, On some problems in combinatorial set theory. Publ. Inst. Math. (Beograd) (N.S.) (1995), 61-65.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2288, "problem_number": "EP-741", "title": "Erdős Problem #741", "statement": "Let $A\\subseteq \\mathbb{N}$ be such that $A+A$ has positive density. Can one always decompose $A=A_1\\sqcup A_2$ such that $A_1+A_1$ and $A_2+A_2$ both have positive density?\nIs there a basis $A$ of order $2$ such that if $A=A_1\\sqcup A_2$ then $A_1+A_1$ and $A_2+A_2$ cannot both have bounded gaps?", "background": "A problem of Burr and Erdos. Erdos \\cite{Er94b} thought he could construct a basis as in the second question, but 'could never quite finish the proof'.\nReferences\n\n\n[Er94b] Erdos, Paul, Some problems in number theory, combinatorics and combinatorial geometry. Math. Pannon. (1994), 261-269.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2289, "problem_number": "EP-749", "title": "Erdős Problem #749", "statement": "Let $\\epsilon>0$. Does there exist $A\\subseteq \\mathbb{N}$ such that the lower density of $A+A$ is at least $1-\\epsilon$ and yet $1_A\\ast 1_A(n) \\ll_\\epsilon 1$ for all $n$?", "background": "A similar question can be asked for upper density.\nSee also [28].\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2290, "problem_number": "EP-750", "title": "Erdős Problem #750", "statement": "Let $f(m)$ be some function such that $f(m)\\to \\infty$ as $m\\to \\infty$. Does there exist a graph $G$ of infinite chromatic number such that every subgraph on $m$ vertices contains an independent set of size at least $\\frac{m}{2}-f(m)$?", "background": "In \\cite{Er69b} Erdos conjectures this for $f(m)=\\epsilon m$ for any fixed $\\epsilon>0$. This follows from a result of Erdos, Hajnal, and Szemer\\'{e}di \\cite{EHS82}, as described by msellke in the comments.\nIn \\cite{ErHa67b} Erdos and Hajnal prove this for $f(m)\\geq cm$ for all $c>1/4$.\nSee also [75].\nReferences\n\n\n[EHS82] Erdos, P. and Hajnal, A. and Szemer\\'{e}di, E., On almost bipartite large chromatic graphs. Theory and practice of combinatorics (1982), 117-123.\n\n[Er69b] Erdos, P., Problems and results in chromatic graph theory. Proof Techniques in Graph Theory (Proc. Second Ann\nArbor Graph Theory Conf., Ann Arbor, Mich.,\n1968) (1969), 27-35.\n\n[ErHa67b] Erdos, P. and Hajnal, Andr\\'as, On chromatic graphs. Mat. Lapok (1967), 1--4.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2291, "problem_number": "EP-757", "title": "Erdős Problem #757", "statement": "Let $A\\subset \\mathbb{R}$ be a set of size $n$ such that every subset $B\\subseteq A$ with $\\lvert B\\rvert =4$ has $\\lvert B-B\\rvert\\geq 11$. Find the best constant $c>0$ such that $A$ must always contain a Sidon set of size $\\geq cn$.", "background": "For comparison, note that if $B$ were a Sidon set then $\\lvert B-B\\rvert=13$, so this condition is saying that at most one difference is 'missing' from $B-B$. Equivalently, one can view $A$ as a set such that every four points determine at least five distinct distances, and ask for a subset with all distances distinct.\nWithout loss of generality, one can assume $A\\subset \\mathbb{N}$.\nErdos and S\\'{o}s proved that $c\\geq 1/2$. Gy\\'{a}rf\\'{a}s and Lehel \\cite{GyLe95} proved $ \\frac{1}{2}1$ such that $d\\equiv 1\\pmod{p}$. Is it true that there exists some constant $c>0$ such that for all large $N$ $ \\frac{\\lvert A\\cap [1,N]\\rvert}{N}=\\exp(-(c+o(1))\\sqrt{\\log N}\\log\\log N). $ ", "background": "Erdos could prove that there exists some constant $c>0$ such that for all large $N$ $ \\exp(-c\\sqrt{\\log N}\\log\\log N)\\leq \\frac{\\lvert A\\cap [1,N]\\rvert}{N} $ and $ \\frac{\\lvert A\\cap [1,N]\\rvert}{N}\\leq \\exp(-(1+o(1))\\sqrt{\\log N\\log\\log N}). $ Erdos asked about this because $\\lvert A\\cap [1,N]\\rvert$ provides an upper bound for the number of integers $n\\leq N$ for which there is a non-cyclic simple group of order $n$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2295, "problem_number": "EP-769", "title": "Erdős Problem #769", "statement": "Let $c(n)$ be minimal such that if $k\\geq c(n)$ then the $n$-dimensional unit cube can be decomposed into $k$ homothetic $n$-dimensional cubes. Give good bounds for $c(n)$ - in particular, is it true that $c(n) \\gg n^n$?", "background": "A problem first investigated by Hadwiger, who proved the lower bound $ c(n) \\geq 2^n+2^{n-1}. $ It is easy to see that $c(2)=6$. Meier conjectured $c(3)=48$. Burgess and Erdos \\cite{Er74b} proved $ c(n) \\ll n^{n+1}. $ Erdos wrote 'I am certain that if $n+1$ is a prime then $c(n)>n^n$.'\nHudelson \\cite{Hu98} proved that if $(2^n-1,3^n-1)=1$ then $c(n) < 6^n$, and in general $c(n) \\ll (2n)^{n-1}$. Connor and Marmorino \\cite{CoMa18} proved that $ c(n) \\geq 2^{n+1}-1 $ for all $n\\geq 3$, $ c(n) \\leq 1.8n^{n+1} $ if $n+1$ is prime, and $ c(n) \\leq e^2n^n $ otherwise.\nReferences\n\n\n[CoMa18] Connor, Peter and Marmorino, Phillip, Decomposing cubes into smaller cubes. J. Geom. (2018), Paper No. 19, 11.\n\n[Er74b] Erdos, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202.\n\n[Hu98] Hudelson, Matthew, Dissecting {$d$}-cubes into smaller {$d$}-cubes. J. Combin. Theory Ser. A (1998), 190--200.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2296, "problem_number": "EP-770", "title": "Erdős Problem #770", "statement": "Let $h(n)$ be minimal such that $2^n-1,3^n-1,\\ldots,h(n)^n-1$ are mutually coprime.\nDoes, for every prime $p$, the density $\\delta_p$ of integers with $h(n)=p$ exist? Does $\\liminf h(n)=\\infty$? Is it true that if $p$ is the greatest prime such that $p-1\\mid n$ and $p>n^\\epsilon$ then $h(n)=p$?", "background": "It is easy to see that $h(n)=n+1$ if and only if $n+1$ is prime, and that $h(n)$ is unbounded for odd $n$.\nIt is probably true that $h(n)=3$ for infinitely many $n$.\nSee also [820].\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2297, "problem_number": "EP-773", "title": "Erdős Problem #773", "statement": "What is the size of the largest Sidon subset $A\\subseteq\\{1,2^2,\\ldots,N^2\\}$? Is it $N^{1-o(1)}$?", "background": "A question of Alon and Erdos \\cite{AlEr85}, who proved $\\lvert A\\rvert \\geq N^{2/3-o(1)}$ is possible (via a random subset), and observed that $ \\lvert A\\rvert \\ll \\frac{N}{(\\log N)^{1/4}}, $ since (as shown by Landau) the density of the sums of two squares decays like $(\\log N)^{-1/2}$. The lower bound was improved to $ \\lvert A\\rvert \\gg N^{2/3} $ by Lefmann and Thiele \\cite{LeTh95}.\nReferences\n\n\n[AlEr85] Alon, Noga and Erdos, P., An application of graph theory to additive number theory. European J. Combin. (1985), 201-203.\n\n[LeTh95] Lefmann, Hanno and Thiele, Torsten, Point sets with distinct distances. Combinatorica (1995), 379--408.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2298, "problem_number": "EP-774", "title": "Erdős Problem #774", "statement": "We call $A\\subset \\mathbb{N}$ dissociated if $\\sum_{n\\in X}n\neq \\sum_{m\\in Y}m$ for all finite $X,Y\\subset A$ with $X\neq Y$.\nLet $A\\subset \\mathbb{N}$ be an infinite set. We call $A$ proportionately dissociated if every finite $B\\subset A$ contains a dissociated set of size $\\gg \\lvert B\\rvert$.\nIs every proportionately dissociated set the union of a finite number of dissociated sets?", "background": "This question appears in a paper of Alon and Erdos \\cite{AlEr85}, although the general topic was first considered by Pisier \\cite{Pi83}, who observed that the converse holds, and proved that being proportionately dissociated is equivalent to being a 'Sidon set' in the harmonic analysis sense; that is, whenever $f:A\\to \\mathbb{C}$ there exists some $\\theta\\in [0,1]$ such that $ \\| f\\|_1 \\ll \\left\\lvert\\sum_{n\\in A} f(n)e(n\\theta)\\right\\rvert, $ where $e(x)=e^{2\\pi ix}$.\nAlon and Erdos write that it 'seems unlikely that [this] is also sufficient'. They also point out the same question can be asked replacing dissociated with Sidon (in the additive combinatorial sense) (see [328]). This latter question was resolved in the negative by Ne\\v{s}et\\v{r}il, R\"{o}dl, and Sales \\cite{NRS24}.\nReferences\n\n\n[AlEr85] Alon, Noga and Erdos, P., An application of graph theory to additive number theory. European J. Combin. (1985), 201-203.\n\n[NRS24] Ne\\v set\\v ril, Jaroslav and R\"odl, Vojt\\v ech and Sales,\nMarcelo, On {P}isier type theorems. Combinatorica (2024), 1211--1232.\n\n[Pi83] Pisier, Gilles, Arithmetic characterizations of Sidon sets. Bull. Amer. Math. Soc. (N.S.) (1983), 87-89.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2299, "problem_number": "EP-776", "title": "Erdős Problem #776", "statement": "Let $r\\geq 2$ and $A_1,\\ldots,A_m\\subseteq \\{1,\\ldots,n\\}$ be such that $A_i\not\\subseteq A_j$ for all $i\neq j$ and for any $t$ if there exists some $i$ with $\\lvert A_i\\rvert=t$ then there must exist at least $r$ sets of that size.\nHow large must $n$ be (as a function of $r$) to ensure that there is such a family which achieves $n-3$ distinct sizes of sets?", "background": "A problem of Erdos and Trotter. For $r=1$ and $n>3$ the maximum possible is $n-2$. For $r>1$ and $n$ sufficiently large $n-3$ is achievable, but $n-2$ is never achievable.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2300, "problem_number": "EP-778", "title": "Erdős Problem #778", "statement": "Alice and Bob play a game on the edges of $K_n$, alternating colouring edges by red (Alice) and blue (Bob). Alice goes first, and wins if at the end the largest red clique is larger than any of the blue cliques.\nDoes Bob have a winning strategy for $n\\geq 3$? (Erdos believed the answer is yes.)", "background": "If we change the game so that Bob colours two edges after each edge that Alice colours, but now require Bob's largest clique to be strictly larger than Alice's, then does Bob have a winning strategy for $n>3$?\nFinally, consider the game when Alice wins if the maximum degree of the red subgraph is larger than the maximum degree of the blue subgraph. Who wins?\nMalekshahian and Spiro \\cite{MaSp24} have proved that, for the first game, the set of $n$ for which Bob wins has density at least $3/4$ - in fact they prove that if Alice wins at $n$ then Bob wins at $n+1,n+2,n+3$.\nSimilarly, for the third game they prove that the set of $n$ for which Bob wins has density at least $2/3$, and prove the stronger statement that if Alice wins at $n$ then Bob wins at $n+1,n+2$.\nReferences\n\n\n[MaSp24] Malekshahian, A. and Spiro, S., On a clique-building game of Erdos. arXiv:2410.18304 (2024).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2301, "problem_number": "EP-782", "title": "Erdős Problem #782", "statement": "Do the squares contain arbitrarily long quasi-progressions? That is, does there exist some constant $C>0$ such that, for any $k$, the squares contain a sequence $x_1,\\ldots,x_k$ where, for some $d$ and all $1\\leq i0$ and let $n$ be large. Let $A\\subseteq \\{2,\\ldots,n\\}$ be such that $(a,b)=1$ for all $a\neq b\\in A$ and $\\sum_{n\\in A}\\frac{1}{n}\\leq C$.\nWhat choice of such an $A$ minimises the number of integers $m\\leq n$ not divisible by any $a\\in A$? Is this minimised by letting $n\\geq q_1>q_2>\\cdots$ be the consecutive primes in decreasing order and choosing $A=\\{q_1,\\ldots,q_k\\}$ where $k$ is maximal such that $ \\sum_{i=1}^k\\frac{1}{q_i}\\leq C? $ \",\n \"difficulty\": \"L1\"\n},{", "background": "", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2303, "problem_number": "EP-786", "title": "Erdős Problem #786", "statement": "Let $\\epsilon>0$. Is there some set $A\\subset \\mathbb{N}$ of density $>1-\\epsilon$ such that $a_1\\cdots a_r=b_1\\cdots b_s$ with $a_i,b_j\\in A$ can only hold when $r=s$?\nSimilarly, can one always find a set $A\\subset\\{1,\\ldots,N\\}$ with this property of size $\\geq (1-o(1))N$?", "background": "An example of such a set with density $1/4$ is given by the integers $\\equiv 2\\pmod{4}$.\nSelfridge constructed such a set with density $1/e-\\epsilon$ for any $\\epsilon>0$: let $p_1<\\cdotsN^{1/2}$ give an example of a set with size $\\geq (\\log 2)N$. Erdos could improve this constant slightly.\nIn \\cite{Er65} Erdos reports that Ruzsa proved the maximal size of such an $A$ is $\\leq (1-c)N$ for some constant $c>0$ for large $N$, but the proof 'is not yet published'. As far as I know, no such proof was ever published.\nSee also [421] and [795].\nReferences\n\n\n[Er65] Erdos, P., Extremal problems in number theory. Proc. Sympos. Pure Math., Vol. VIII (1965), 181-189.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2304, "problem_number": "EP-787", "title": "Erdős Problem #787", "statement": "Let $g(n)$ be maximal such that given any set $A\\subset \\mathbb{R}$ with $\\lvert A\\rvert=n$ there exists some $B\\subseteq A$ of size $\\lvert B\\rvert\\geq g(n)$ such that $b_1+b_2\not\\in A$ for all $b_1\neq b_2\\in B$.\nEstimate $g(n)$.", "background": "This function was considered by Erdos and Moser. Choi observed that, without loss of generality, one can assume that $A\\subset \\mathbb{Z}$.\nKlarner proved $g(n) \\gg \\log n$ (indeed, a greedy construction suffices). Choi \\cite{Ch71} proved $g(n) \\ll n^{2/5+o(1)}$. The current best bounds known are $ (\\log n)^{1+c} \\ll g(n) \\ll \\exp(\\sqrt{\\log n}) $ for some constant $c>0$, the lower bound due to Sanders \\cite{Sa21} and the upper bound due to Ruzsa \\cite{Ru05}. Beker \\cite{Be25} has proved $ (\\log n)^{1+\\tfrac{1}{68}+o(1)} \\ll g(n). $ \nReferences\n\n\n[Be25] A. Beker, The Erdos-Moser sum-free set problem via improved bounds for $k$-configurations. arXiv:2501.10203 (2025).\n\n[Ch71] Choi, S. L. G., On a combinatorial problem in number theory. Proc. London Math. Soc. (3) (1971), 629-642.\n\n[Ru05] Ruzsa, Imre Z., Sum-avoiding subsets. Ramanujan J. (2005), 77-82.\n\n[Sa21] Sanders, Tom, The Erdos-Moser sum-free set problem. Canad. J. Math. (2021), 63-107.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2305, "problem_number": "EP-788", "title": "Erdős Problem #788", "statement": "Let $f(n)$ be maximal such that if $B\\subset (2n,4n)\\cap \\mathbb{N}$ there exists some $C\\subset (n,2n)\\cap \\mathbb{N}$ such that $c_1+c_2\not\\in B$ for all $c_1\neq c_2\\in C$ and $\\lvert C\\rvert+\\lvert B\\rvert \\geq f(n)$.\nEstimate $f(n)$. In particular is it true that $f(n)\\leq n^{1/2+o(1)}$?", "background": "A conjecture of Choi \\cite{Ch71}, who proved $f(n) \\ll n^{3/4}$. Adenwalla in the comments has provided a simple construction that proves $f(n) \\gg n^{1/2}$.\nHunter in the comments has sketched an argument that gives $f(n) \\ll n^{2/3+o(1)}$. The bound $ f(n) \\ll (n\\log n)^{2/3} $ was proved by Baltz, Schoen, and Srivastav \\cite{BSS00}.\nReferences\n\n\n[BSS00] Baltz, Andreas and Schoen, Tomasz and Srivastav, Anand, Probabilistic construction of small strongly sum-free sets via\nlarge {S}idon sets. Colloq. Math. (2000), 171--176.\n\n[Ch71] Choi, S. L. G., On a combinatorial problem in number theory. Proc. London Math. Soc. (3) (1971), 629-642.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2306, "problem_number": "EP-789", "title": "Erdős Problem #789", "statement": "Let $h(n)$ be maximal such that if $A\\subseteq \\mathbb{Z}$ with $\\lvert A\\rvert=n$ then there is $B\\subseteq A$ with $\\lvert B\\rvert \\geq h(n)$ such that if $a_1+\\cdots+a_r=b_1+\\cdots+b_s$ with $a_i,b_i\\in B$ then $r=s$.\nEstimate $h(n)$.", "background": "Erdos \\cite{Er62c} proved $h(n) \\ll n^{5/6}$. Straus \\cite{St66} proved $h(n) \\ll n^{1/2}$. Erdos noted the bound $h(n)\\gg n^{1/3}$, taking $ B=\\{ a: \\{ \\alpha a\\} \\in n^{-1/3}+\\tfrac{1}{2} (-n^{-2/3},n^{-2/3})\\} $ for a random $\\alpha\\in [0,1]$. \\cite{Er62c} and Choi \\cite{Ch74b} improved this to $h(n) \\gg (n\\log n)^{1/3}$.\nSee also [186] and [874].\nReferences\n\n\n[Ch74b] Choi, S. L. G., On an extremal problem in number theory. J. Number Theory (1974), 105--111.\n\n[Er62c] Erdos, P\\'{a}l, Some remarks on number theory. {III}. Mat. Lapok (1962), 28--38.\n\n[St66] Straus, E. G., On a problem in combinatorial number theory. J. Math. Sci. (1966), 77--80.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2307, "problem_number": "EP-790", "title": "Erdős Problem #790", "statement": "Let $l(n)$ be maximal such that if $A\\subset\\mathbb{Z}$ with $\\lvert A\\rvert=n$ then there exists a sum-free $B\\subseteq A$ with $\\lvert B\\rvert \\geq l(n)$ - that is, $B$ is such that there are no solutions to $ a_1=a_2+\\cdots+a_r $ with $a_i\\in B$ all distinct.\nEstimate $l(n)$. In particular, is it true that $l(n)n^{-1/2}\\to \\infty$? Is it true that $l(n)< n^{1-c}$ for some $c>0$?", "background": "Erdos observed that $l(n)\\geq (n/2)^{1/2}$, which Choi improved to $l(n)>(1+c)n^{1/2}$ for some $c>0$. Erdos \\cite{Er73} thought he could prove $l(n)=o(n)$ but had 'difficulties in reconstructing [his] proof'. (In \\cite{Er65} he wrote 'by complicated arguments we can show $l(n)=o(n)$'.)\nChoi, Koml\\'{o}s, and Szemer\\'{e}di \\cite{CKS75} proved $ \\left(\\frac{\\log n}{\\log\\log n}n\\right)^{1/2}\\ll l(n) \\ll \\frac{n}{\\log n}. $ They further conjecture that $l(n)\\geq n^{1-o(1)}$.\nSee also [876].\nReferences\n\n\n[CKS75] Choi, S. L. G. and Koml\\'os, J. and Szemer\\'{e}di, E., On sum-free subsequences. Trans. Amer. Math. Soc. (1975), 307--313.\n\n[Er65] Erdos, P., Extremal problems in number theory. Proc. Sympos. Pure Math., Vol. VIII (1965), 181-189.\n\n[Er73] Erdos, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2308, "problem_number": "EP-791", "title": "Erdős Problem #791", "statement": "Let $g(n)$ be minimal such that there exists $A\\subseteq \\{0,\\ldots,n\\}$ of size $g(n)$ with $\\{0,\\ldots,n\\}\\subseteq A+A$. Estimate $g(n)$. In particular is it true that $g(n)\\sim 2n^{1/2}$?", "background": "Such a set is often called a finite additive $2$-basis. A problem of Rohrbach, who proved in \\cite{Ro37} $ (2+c)n \\leq g(n)^2 \\leq 4n $ for some small constant $c>0$. The current best-known bounds are $ (2.181\\cdots+o(1))n\\leq g(n)^2 \\leq (3.458\\cdots+o(1))n. $ The lower bound is due to Yu \\cite{Yu15}, and the upper bound is due to Kohonen \\cite{Ko17}. (The disproof of $g(n)\\sim 2n^{1/2}$ was accomplished by Mrose \\cite{Mr79}, who gave a construction implying $g(n)^2 \\leq \\frac{7}{2}n$.)\nReferences\n\n\n[Ko17] Kohonen, Jukka, An improved lower bound for finite additive 2-bases. J. Number Theory (2017), 518--524.\n\n[Mr79] Mrose, Arnulf, Untere {S}chranken f\"ur die {R}eichweiten von {E}xtremalbasen\nfester {O}rdnung. Abh. Math. Sem. Univ. Hamburg (1979), 118--124.\n\n[Ro37] Rohrbach, Hans, Ein {B}eitrag zur additiven {Z}ahlentheorie. Math. Z. (1937), 1--30.\n\n[Yu15] Yu, Gang, A new upper bound for finite additive {$h$}-bases. J. Number Theory (2015), 95--104.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2309, "problem_number": "EP-792", "title": "Erdős Problem #792", "statement": "Let $f(n)$ be maximal such that in any $A\\subset \\mathbb{Z}$ with $\\lvert A\\rvert=n$ there exists some sum-free subset $B\\subseteq A$ with $\\lvert B\\rvert \\geq f(n)$, so that there are no solutions to $ a+b=c $ with $a,b,c\\in B$. Estimate $f(n)$.", "background": "Erdos \\cite{Er65} gave a simple proof that shows $f(n) \\geq n/3$. Alon and Kleitman \\cite{AlKl90} improved this to $f(n)\\geq \\frac{n+1}{3}$, and Bourgain \\cite{Bo97} further improved this to $\\frac{n+2}{3}$. The best lower bound known is $ f(n)\\geq \\frac{n}{3}+c\\log\\log n $ for some constant $c>0$, due to Bedert \\cite{Be25b}. The best upper bound known is $ f(n) \\leq \\frac{n}{3}+o(n), $ due to Eberhard, Green, and Manners \\cite{EGM14}.\nThis problem is Problem 1 on Green's open problems list.\nReferences\n\n\n[AlKl90] Alon, N. and Kleitman, D. J., Sum-free subsets. (1990), 13--26.\n\n[Be25b] B. Bedert, Large sum-free subsets of sets of integers via $L^1$-estimates for trigonometric sums. arXiv:2502.08624 (2025).\n\n[Bo97] Bourgain, Jean, Estimates related to sumfree subsets of sets of integers. Israel J. Math. (1997), 71-92.\n\n[EGM14] Eberhard, Sean and Green, Ben and Manners, Freddie, Sets of integers with no large sum-free subset. Ann. of Math. (2) (2014), 621-652.\n\n[Er65] Erdos, P., Extremal problems in number theory. Proc. Sympos. Pure Math., Vol. VIII (1965), 181-189.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2310, "problem_number": "EP-793", "title": "Erdős Problem #793", "statement": "Let $F(n)$ be the maximum possible size of a subset $A\\subseteq\\{1,\\ldots,n\\}$ such that $a\nmid bc$ whenever $a,b,c\\in A$ with $a\neq b$ and $a\neq c$. Is there a constant $C$ such that $ F(n)=\\pi(n)+(C+o(1))n^{2/3}(\\log n)^{-2}? $ ", "background": "Erdos \\cite{Er38} proved there exist constants $0g(n)\\geq (\\log n)^2$ is there a graph on $n$ vertices in which every induced subgraph on $g(n)$ vertices contains a clique of size $\\geq \\log n$ and an independent set of size $\\geq \\log n$?\nIn particular, is there such a graph for $g(n)=(\\log n)^3$?", "background": "A problem of Erdos and Hajnal, who thought that there is no such graph for $g(n)=(\\log n)^3$. Alon and Sudakov \\cite{AlSu07} proved that there is no such graph with $ g(n)=\\frac{c}{\\log\\log n}(\\log n)^3 $ for some constant $c>0$.\nAlon, Buci\\'{c}, and Sudakov \\cite{ABS21} construct such a graph with $ g(n)\\leq 2^{2^{(\\log\\log n)^{1/2+o(1)}}}. $ See also [804].\nReferences\n\n\n[ABS21] Alon, Noga and Buci\\'c, Matija and Sudakov, Benny, Large cliques and independent sets all over the place. Proc. Amer. Math. Soc. (2021), 3145-3157.\n\n[AlSu07] Alon, Noga and Sudakov, Benny, On graphs with subgraphs having large independence numbers. J. Graph Theory (2007), 149-157.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2314, "problem_number": "EP-809", "title": "Erdős Problem #809", "statement": "Let $k\\geq 3$ and define $F_k(n)$ to be the minimal $r$ such that there is a graph $G$ on $n$ vertices with $\\lfloor n^2/4\\rfloor+1$ many edges such that the edges can be $r$-coloured so that every subgraph isomorphic to $C_{2k+1}$ has no colour repeating on the edges.\nIs it true that $ F_k(n)\\sim n^2/8? $ ", "background": "A problem of Burr, Erdos, Graham, and S\\'{o}s, who proved that $ F_k(n)\\gg n^2. $ See also [810].\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2315, "problem_number": "EP-810", "title": "Erdős Problem #810", "statement": "Does there exist some $\\epsilon>0$ such that, for all sufficiently large $n$, there exists a graph $G$ on $n$ vertices with at least $\\epsilon n^2$ many edges such that the edges can be coloured with $n$ colours so that every $C_4$ receives $4$ distinct colours?", "background": "A problem of Burr, Erdos, Graham, and S\\'{o}s.\nSee also [809].\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2316, "problem_number": "EP-811", "title": "Erdős Problem #811", "statement": "Suppose $n\\equiv 1\\pmod{m}$. We say that an edge-colouring of $K_n$ using $m$ colours is balanced if every vertex sees exactly $\\lfloor n/m\\rfloor$ many edges of each colours.\nFor which graphs $G$ is it true that, if $m=e(G)$, for all large $n\\equiv 1\\pmod{m}$, every balanced edge-colouring of $K_n$ with $m$ colours contains a rainbow copy of $G$? (That is, a subgraph isomorphic to $G$ where each edge receives a different colour.)", "background": "In \\cite{Er91} Erdos credits this problem to himself, Pyber, and Tuza. This problem was explored in a paper of Erdos and Tuza \\cite{ErTu93}. In \\cite{Er96} Erdos seems to suggest that this might be true for every graph $G$, and specifically asks specific challenge posed in \\cite{Er91} and \\cite{Er96} is whether, in any balanced edge-colouring of $K_{6n+1}$ by $6$ colours there must exist a rainbow $C_6$ and $K_4$.\nIn general, one can ask for a quantitative version, defining $d_G(n)$ to be minimal (if it exists) such that if $n$ is sufficiently large and the edges of $K_n$ are coloured with $e(G)$ many colours such that the minimum degree of each colour class is $\\geq d_G(n)$ then there is a rainbow copy of $G$. Erdos and Tuza \\cite{ErTu93} proved that $ \\lfloor n/6\\rfloor \\leq d_{C_4}(n) \\leq \\left(\\frac{1}{4}-c\\right)n $ for some constant $c>0$.\nAxenovich and Clemen \\cite{AxCl24} have proved that there exist infinitely many graphs without this property. In particular, they show that for any odd $\\ell \\geq 3$ and $m=\\lfloor \\sqrt{\\ell}+3.5\\rfloor$ there exist arbitrarily large $n$ such that $K_n$ has a balanced edge-colouring using $\\ell$ colours which contains no rainbow $K_m$. They conjecture that $K_m$ lacks this property for all $m\\geq 4$.\nClemen and Wagner \\cite{ClWa23} proved that $K_4$ does lack this property.\nReferences\n\n\n[AxCl24] Axenovich, Maria and Clemen, Felix C., Rainbow subgraphs in edge-colored complete graphs: answering\ntwo questions by {E}rd\\H{o}s and {T}uza. J. Graph Theory (2024), 57--66.\n\n[ClWa23] Clemen, Felix Christian and Wagner, Adam Zsolt, Balanced edge-colorings avoiding rainbow cliques of size four. Electron. J. Combin. (2023), Paper No. 3.17, 3.\n\n[Er91] Erd\"{o}s, P., Problems and results in combinatorial analysis and combinatorial number theory. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988) (1991), 397-406.\n\n[Er96] Erdos, Paul, Some of my favourite problems on cycles and colourings. Tatra Mt. Math. Publ. (1996), 7-9.\n\n[ErTu93] Erdos, Paul and Tuza, Zsolt, Rainbow subgraphs in edge-colorings of complete graphs. (1993), 81--88.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2317, "problem_number": "EP-812", "title": "Erdős Problem #812", "statement": "Is it true that $ \\frac{R(n+1)}{R(n)}\\geq 1+c $ for some constant $c>0$, for all large $n$? Is it true that $ R(n+1)-R(n) \\gg n^2? $ ", "background": "Burr, Erdos, Faudree, and Schelp \\cite{BEFS89} proved that $ R(n+1)-R(n) \\geq 4n-8 $ for all $n\\geq 2$. The lower bound of [165] implies that $ R(n+2)-R(n) \\gg n^{2-o(1)}. $ \nReferences\n\n\n[BEFS89] Burr, S. A. and Erdos, P. and Faudree, R. J. and Schelp, R.\nH., On the difference between consecutive {R}amsey numbers. Utilitas Math. (1989), 115--118.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2318, "problem_number": "EP-813", "title": "Erdős Problem #813", "statement": "Let $h(n)$ be minimal such that every graph on $n$ vertices where every set of $7$ vertices contains a triangle (a copy of $K_3$) must contain a clique on at least $h(n)$ vertices. Estimate $h(n)$ - in particular, do there exist constants $c_1,c_2>0$ such that $ n^{1/3+c_1}\\ll h(n) \\ll n^{1/2-c_2}? $ ", "background": "A problem of Erdos and Hajnal, who could prove that $ n^{1/3}\\ll h(n) \\ll n^{1/2}. $ Buci\\'{c} and Sudakov \\cite{BuSu23} have proved $ h(n) \\gg n^{5/12-o(1)}. $ \nReferences\n\n\n[BuSu23] M. Buci\\'C and B. Sudakov, Large independent sets from local considerations. arXiv:2007.03667 (2023).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2319, "problem_number": "EP-817", "title": "Erdős Problem #817", "statement": "Let $k\\geq 3$ and define $g_k(n)$ to be the minimal $N$ such that $\\{1,\\ldots,N\\}$ contains some $A$ of size $\\lvert A\\rvert=n$ such that $ \\langle A\\rangle = \\left\\{\\sum_{a\\in A}\\epsilon_aa: \\epsilon_a\\in \\{0,1\\}\\right\\} $ contains no non-trivial $k$-term arithmetic progression. Estimate $g_k(n)$. In particular, is it true that $ g_3(n) \\gg 3^n? $ ", "background": "A problem of Erdos and S\\'{a}rk\"{o}zy who proved $ g_3(n) \\gg \\frac{3^n}{n^{O(1)}}. $ \",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2320, "problem_number": "EP-819", "title": "Erdős Problem #819", "statement": "Let $f(N)$ be maximal such that there exists $A\\subseteq \\{1,\\ldots,N\\}$ with $\\lvert A\\rvert=\\lfloor N^{1/2}\\rfloor$ such that $\\lvert (A+A)\\cap [1,N]\\rvert=f(N)$. Estimate $f(N)$.", "background": "Erdos and Freud \\cite{ErFr91} proved $ \\left(\\frac{3}{8}-o(1)\\right)N \\leq f(N) \\leq \\left(\\frac{1}{2}+o(1)\\right)N, $ and note that it is closely connected to the size of the largest quasi-Sidon set (see [840]).\nReferences\n\n\n[ErFr91] Erdos, P. and Freud, R., On sums of a {S}idon-sequence. J. Number Theory (1991), 196--205.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2321, "problem_number": "EP-820", "title": "Erdős Problem #820", "statement": "Let $H(n)$ be the smallest integer $l$ such that there exist $k0$ such that, for all $\\epsilon>0$, $ H(n) > \\exp(n^{(c-\\epsilon)/\\log\\log n}) $ for infinitely many $n$ and $ H(n) < \\exp(n^{(c+\\epsilon)/\\log\\log n}) $ for all large enough $n$?\nDoes a similar upper bound hold for the smallest $k$ such that $(k^n-1,2^n-1)=1$?", "background": "Erdos \\cite{Er74b} proved that there exists a constant $c>0$ such that $ H(n) > \\exp(n^{c/(\\log\\log n)^2}) $ for infinitely many $n$.\nvan Doorn in the comments sketches a proof of the lower bound: that there exists some constant $c>0$ and infinitely many $n$ such that $ H(n) > \\exp(n^{c/\\log\\log n}). $ The sequence $H(n)$ for $1\\leq n\\leq 10$ is $ 3,3,3,6,3,18,3,6,3,12. $ The sequence of $n$ for which $(2^n-1,3^n-1)=1$ is A263647 in the OEIS.\nSee also [770].\nReferences\n\n\n[Er74b] Erdos, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2322, "problem_number": "EP-821", "title": "Erdős Problem #821", "statement": "Let $g(n)$ count the number of $m$ such that $\\phi(m)=n$. Is it true that, for every $\\epsilon>0$, there exist infinitely many $n$ such that $ g(n) > n^{1-\\epsilon}? $ ", "background": "Pillai proved that $\\limsup g(n)=\\infty$ and Erdos \\cite{Er35b} proved that there exists some constant $c>0$ such that $g(n) >n^c$ for infinitely many $n$.\nThis conjecture would follow if we knew that, for every $\\epsilon>0$, there are $\\gg_\\epsilon \\frac{x}{\\log x}$ many primes $p n^{0.71568\\cdots}, $ obtained by Lichtman \\cite{Li22} as a consequence of proving that there are $\\geq \\frac{x}{(\\log x)^{O(1)}}$ many primes $p\\leq x$ such that all prime factors of $p-1$ are $\\leq x^{0.2843\\cdots}$ (which improves a number of previous exponents, most recently Baker and Harman \\cite{BaHa98}).\nThe average size of $g(n)$ was investigated by Luca and Pollack \\cite{LuPo11}.\nSee also [416].\nReferences\n\n\n[BaHa98] Baker, R. C. and Harman, G., Shifted primes without large prime factors. Acta Arith. (1998), 331--361.\n\n[Er35b] Erdos, P., On the normal number of prime factors of $p-1$ and some related problems concerning Euler's $\\varphi$-function. Quart. J. Math. (1935), 205-213.\n\n[Li22] J. D. Lichtman, Primes in arithmetic progressions to large moduli and shifted primes without large prime factors. arXiv:2211.09641 (2022).\n\n[LuPo11] Luca, Florian and Pollack, Paul, An arithmetic function arising from {C}armichael's conjecture. J. Th\\'{e}or. Nombres Bordeaux (2011), 697--714.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2323, "problem_number": "EP-824", "title": "Erdős Problem #824", "statement": "Let $h(x)$ count the number of integers $1\\leq ax^{2-o(1)}$?", "background": "Erdos \\cite{Er74b} proved that $\\limsup h(x)/x= \\infty$, and claimed a similar proof for this problem. A complete proof that $h(x)/x\\to \\infty$ was provided by Pollack and Pomerance \\cite{PoPo16}.\nA similar question can be asked if we replace the condition $(a,b)=1$ with the condition that $a$ and $b$ are squarefree. Weisenberg suggests another variant, with the condition that there are no proper factors $u\\mid a$ and $v\\mid b$ such that $\\sigma(u)=\\sigma(v)$ and $(u,a/u)=(v,b/v)=1$, which is the weakest restriction one can impose that is still strong enough to eliminate trivial duplicates.\nReferences\n\n\n[Er74b] Erdos, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202.\n\n[PoPo16] Pollack, Paul and Pomerance, Carl, Some problems of Erdos on the sum-of-divisors function. Trans. Amer. Math. Soc. Ser. B (2016), 1-26.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2324, "problem_number": "EP-825", "title": "Erdős Problem #825", "statement": "Is there an absolute constant $C>0$ such that every integer $n$ with $\\sigma(n)>Cn$ is the distinct sum of proper divisors of $n$?", "background": "A problem of Benkoski and Erdos. In other words, this problem asks for an upper bound for the abundancy index of weird numbers. This could be true with $C=3$. We must have $C>2$ since $\\sigma(70)=144$ but $70$ is not the distinct sum of integers from $\\{1,2,5,7,10,14,35\\}$.\nErdos suggested that as $C\\to \\infty$ only divisors at most $\\epsilon n$ need to be used, where $\\epsilon \\to 0$.\nWeisenberg has observed that if $n$ is a weird number with an abundancy index $\\geq 4$ then it is divisible by an odd weird number. In particular, if there are no odd weird numbers (see [470]) then every weird number has abundancy index $<4$. Indeed, if $l(n)$ is the abundancy index and $n=2^km$ with $m$ odd then $l(n)=l(2^k)l(m)$, and $l(2^k)<2$ so if $l(n)\\geq 4$ then $l(m)>2$, and hence $m$ is weird (as a factor of a weird number).\nA similar argument shows that either there are infinitely many primitive weird numbers or there is an upper bound for the abundancy index of all weird numbers.\nSee also [18] and [470].\nThis is part of problem B2 in Guy's collection \\cite{Gu04} (the \\$25 is reported by Guy as offered by Erdos for a solution to this question).\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2325, "problem_number": "EP-826", "title": "Erdős Problem #826", "statement": "Are there infinitely many $n$ such that, for all $k\\geq 1$, $ \\tau(n+k)\\ll k? $ ", "background": "A stronger form of [248].\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2326, "problem_number": "EP-827", "title": "Erdős Problem #827", "statement": "Let $n_k$ be minimal such that if $n_k$ points in $\\mathbb{R}^2$ are in general position then there exists a subset of $k$ points such that all $\\binom{k}{3}$ triples determine circles of different radii.\nDetermine $n_k$.", "background": "In \\cite{Er75h} Erdos asks whether $n_k$ exists. In \\cite{Er78c} he gave a simple argument which proves that it does, and in fact $ n_k \\leq k+2\\binom{k-1}{2}\\binom{k-1}{3}, $ but this argument is incorrect, as explained by Martinez and Rold\\'{a}n-Pensado \\cite{MaRo15}.\nMartinez and Rold\\'{a}n-Pensado give a corrected argument that proves $n_k\\ll k^9$.\nReferences\n\n\n[Er75h] Erdos, P., Some problems on elementary geometry. Austral. Math. Soc. Gaz. (1975), 2-3.\n\n[Er78c] Erdos, P., Some more problems on elementary geometry. Austral. Math. Soc. Gaz. (1978), 52-54.\n\n[MaRo15] Mart\\'{I}nez, L. and Rold\\'an-Pensado, E., Points defining triangles with distinct circumradii. Acta Math. Hungar. (2015), 136--141.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2327, "problem_number": "EP-828", "title": "Erdős Problem #828", "statement": "Is it true that, for any $a\\in\\mathbb{Z}$, there are infinitely many $n$ such that $ \\phi(n) \\mid n+a? $ ", "background": "A conjecture of Graham. Lehmer has conjectured that $\\phi(n)\\mid n-1$ if and only if $n$ is prime. It is an easy exercise to show that $\\phi(n) \\mid n$ if and only if $n=2^a3^b$.\nThis is discussed in problem B37 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2328, "problem_number": "EP-829", "title": "Erdős Problem #829", "statement": "Let $A\\subset\\mathbb{N}$ be the set of cubes. Is it true that $ 1_A\\ast 1_A(n) \\ll (\\log n)^{O(1)}? $ ", "background": "Mordell proved that $ \\limsup_{n\\to \\infty} 1_A\\ast 1_A(n)=\\infty $ and Mahler \\cite{Ma35b} proved $ 1_A\\ast 1_A(n) \\gg (\\log n)^{1/4} $ for infinitely many $n$. Stewart \\cite{St08} improved this to $ 1_A\\ast 1_A(n) \\gg (\\log n)^{11/13}. $ \nReferences\n\n\n[Ma35b] Mahler, Kurt, On the Lattice Points on Curves of Genus 1. Proc. London Math. Soc. (2) (1935), 431-466.\n\n[St08] Stewart, Cameron L., Cubic {T}hue equations with many solutions. Int. Math. Res. Not. IMRN (2008), Art. ID rnn040, 11.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2329, "problem_number": "EP-830", "title": "Erdős Problem #830", "statement": "We say that $a,b\\in \\mathbb{N}$ are an amicable pair if $\\sigma(a)=\\sigma(b)=a+b$. Are there infinitely many amicable pairs? If $A(x)$ counts the number of amicable $1\\leq a\\leq b\\leq x$ then is it true that $ A(x)>x^{1-o(1)}? $ ", "background": "For example $220$ and $284$. Erdos \\cite{Er55b} proved that $A(x)=o(x)$, and Pomerance \\cite{Po81} improved this to $ A(x) \\leq x \\exp(-(\\log x)^{1/3}) $ and later \\cite{Po15} to $ A(x) \\leq x \\exp(-(\\tfrac{1}{2}+o(1))(\\log x\\log\\log x)^{1/2}). $ This is problem B4 in Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Er55b] Erd\"{o}s, P., On amicable numbers. Publ. Math. Debrecen (1955), 108-111.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Po15] Pomerance, Carl, On amicable numbers. (2015), 321-327.\n\n[Po81] Pomerance, Carl, On the distribution of amicable numbers. {II}. J. Reine Angew. Math. (1981), 183-188.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2330, "problem_number": "EP-831", "title": "Erdős Problem #831", "statement": "Let $h(n)$ be maximal such that in any $n$ points in $\\mathbb{R}^2$ (with no three on a line and no four on a circle) there are at least $h(n)$ many circles of different radii passing through three points. Estimate $h(n)$.\n\",\n \"difficulty\": \"L1\"\n},{", "background": "", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2331, "problem_number": "EP-836", "title": "Erdős Problem #836", "statement": "Let $r\\geq 2$ and $G$ be a $r$-uniform hypergraph with chromatic number $3$ (that is, there is a $3$-colouring of the vertices of $G$ such that no edge is monochromatic).\nSuppose any two edges of $G$ have a non-empty intersection. Must $G$ contain $O(r^2)$ many vertices? Must there be two edges which meet in $\\gg r$ many vertices?", "background": "A problem of Erdos and Shelah. The Fano geometry gives an example where there are no two edges which meet in $r-1$ vertices. Are there any other examples?\nErdos and Lov\\'{a}sz \\cite{ErLo75} proved that there must be two edges which meet in $\\gg \\frac{r}{\\log r}$ many vertices.\nAlon has provided the following counterexample to the first question: as vertices take two sets $X$ and $Y$ of sizes $2r-2$ and $\\frac{1}{2}\\binom{2r-2}{r-1}$ respectively, where $Y$ corresponds to all partitions of $X$ into two equal parts. The edges are all subsets of $X$ of size $r$, and also all sets consisting of a subset of $X$ of size $r-1$ together with the unique element of $Y$ corresponding to the induced partition of $X$.\nThis hypergraph is intersecting, its chromatic number is $3$, and it has $\\asymp 4^r/\\sqrt{r}$ many vertices.\nReferences\n\n\n[ErLo75] Erdos, P. and Lov\\'{a}sz, L., Problems and results on {$3$}-chromatic hypergraphs and some\nrelated questions. (1975), 609--627.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2332, "problem_number": "EP-837", "title": "Erdős Problem #837", "statement": "Let $k\\geq 2$ and $A_k\\subseteq [0,1]$ be the set of $\\alpha$ such that there exists some $\\beta(\\alpha)>\\alpha$ with the property that, if $G_1,G_2,\\ldots$ is a sequence of $k$-uniform hypergraphs with $ \\liminf \\frac{e(G_n)}{\\binom{\\lvert G_n\\rvert}{k}} >\\alpha $ then there exist subgraphs $H_n\\subseteq G_n$ such that $\\lvert H_n\\rvert \\to \\infty$ and $ \\liminf \\frac{e(H_n)}{\\binom{\\lvert H_n\\rvert}{k}} >\\beta, $ and further that this property does not necessarily hold if $>\\alpha$ is replaced by $\\geq \\alpha$.\nWhat is $A_3$?", "background": "A problem of Erdos and Simonovits. It is known that $ A_2 = \\left\\{ 1-\\frac{1}{k} : k\\geq 1\\right\\}. $ \n\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2333, "problem_number": "EP-838", "title": "Erdős Problem #838", "statement": "Let $f(n)$ be maximal such that any $n$ points in $\\mathbb{R}^2$, with no three on a line, determine at least $f(n)$ different convex subsets. Estimate $f(n)$ - in particular, does there exist a constant $c$ such that $ \\lim \\frac{\\log f(n)}{(\\log n)^2}=c? $ ", "background": "A question of Erdos and Hammer. Erdos proved in \\cite{Er78c} that there exist constants $c_1,c_2>0$ such that $ n^{c_1\\log n}1/2$. In fact this is false - Freud \\cite{Fr93} constructed a sequence with upper density $19/36$.\nSee also [359] and [867].\nReferences\n\n\n[Fr93] R. Freud, Adding numbers - on a problem of P. Erdos. James Cook Mathematical Notes (1993), 6199-6202.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2335, "problem_number": "EP-840", "title": "Erdős Problem #840", "statement": "Let $f(N)$ be the size of the largest quasi-Sidon subset $A\\subset\\{1,\\ldots,N\\}$, where we say that $A$ is quasi-Sidon if $ \\lvert A+A\\rvert=(1+o(1))\\binom{\\lvert A\\rvert}{2}. $ How does $f(N)$ grow?", "background": "Considered by Erdos and Freud \\cite{ErFr91}, who proved $ \\left(\\frac{2}{\\sqrt{3}}+o(1)\\right)N^{1/2} \\leq f(N) \\leq \\left(2+o(1)\\right)N^{1/2}. $ (Although both bounds were already given by Erdos in \\cite{Er81h}.) Note that $2/\\sqrt{3}=1.15\\cdots$. The lower bound is taking a genuine Sidon set $B\\subset [1,N/3]$ of size $\\sim N^{1/2}/\\sqrt{3}$ and taking the union with $\\{N-b : b\\in B\\}$. The upper bound was improved by Pikhurko \\cite{Pi06} to $ f(N) \\leq \\left(\\left(\\frac{1}{4}+\\frac{1}{(\\pi+2)^2}\\right)^{-1/2}+o(1)\\right)N^{1/2} $ (the constant here is $=1.863\\cdots$).\nThe analogous question with $A-A$ in place of $A+A$ is simpler, and there the maximal size is $\\sim N^{1/2}$, as proved by Cilleruelo.\nSee also [30], [819], and [864].\nReferences\n\n\n[Er81h] Erdos, P., Some problems and results on additive and multiplicative\nnumber theory. Analytic number theory (Philadelphia, Pa., 1980) (1981), 171-182.\n\n[ErFr91] Erdos, P. and Freud, R., On sums of a {S}idon-sequence. J. Number Theory (1991), 196--205.\n\n[Pi06] Pikhurko, Oleg, Dense edge-magic graphs and thin additive bases. Discrete Math. (2006), 2097--2107.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2336, "problem_number": "EP-846", "title": "Erdős Problem #846", "statement": "Let $A\\subset \\mathbb{R}^2$ be an infinite set for which there exists some $\\epsilon>0$ such that in any subset of $A$ of size $n$ there are always at least $\\epsilon n$ with no three on a line.\nIs it true that $A$ is the union of a finite number of sets where no three are on a line?", "background": "A problem of Erdos, Ne\\v{s}et\\v{r}il, and R\"{o}dl.\nSee also [774] and [847].\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2337, "problem_number": "EP-847", "title": "Erdős Problem #847", "statement": "Let $A\\subset \\mathbb{N}$ be an infinite set for which there exists some $\\epsilon>0$ such that in any subset of $A$ of size $n$ there is a subset of size at least $\\epsilon n$ which contains no three-term arithmetic progression.\nIs it true that $A$ is the union of a finite number of sets which contain no three-term arithmetic progression?", "background": "A problem of Erdos, Ne\\v{s}et\\v{r}il, and R\"{o}dl.\nSee also [774] and [846].\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2338, "problem_number": "EP-849", "title": "Erdős Problem #849", "statement": "Is it true that, for every integer $t\\geq 1$, there is some integer $a$ such that $ \\binom{n}{k}=a $ (with $1\\leq k\\leq n/2$) has exactly $t$ solutions?", "background": "Erdos \\cite{Er96b} credits this to himself and Gordon 'many years ago', but it is more commonly known as Singmaster's conjecture. For $t=3$ one could take $a=120$, and for $t=4$ one could take $a=3003$. There are no known examples for $t\\geq 5$.\nBoth Erdos and Singmaster believed the answer to this question is no, and in fact that there exists an absolute upper bound on the number of solutions.\nMatomäki, Radziwill, Shao, Tao, and Teräväinen \\cite{MRSTT22} have proved that there are always at most two solutions if we restrict $k$ to $ k\\geq \\exp((\\log n)^{2/3+\\epsilon}), $ assuming $a$ is sufficiently large depending on $\\epsilon>0$.\nReferences\n\n\n[Er96b] Erd\"{o}s, Paul, Some problems I presented or planned to present in my short\ntalk. Analytic number theory, Vol. 1 (Allerton Park, IL, 1995) (1996), 333-335.\n\n[MRSTT22] Matom\"{a}ki, Kaisa and Radziwi\\l\\l, Maksym and Shao, Xuancheng\nand Tao, Terence and Ter\"{a}v\"{a}inen, Joni, Singmaster's conjecture in the interior of {P}ascal's\ntriangle. Q. J. Math. (2022), 1137--1177.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2339, "problem_number": "EP-850", "title": "Erdős Problem #850", "statement": "Can there exist two distinct integers $x$ and $y$ such that $x,y$ have the same prime factors, $x+1,y+1$ have the same prime factors, and $x+2,y+2$ also have the same prime factors?", "background": "This is sometimes known as the Erdos-Woods conjecture.\nFor just $x,y$ and $x+1,y+1$ one can take $ x=2(2^r-1) $ and $ y = x(x+2). $ Erdos also asked whether there are any other examples. Makowski \\cite{Ma68} observed that $x=75$ and $y=1215$ is another example, since $ 75 = 3\\cdot 5^2 \\textrm{ and }1215 = 3^5\\cdot 5 $ while $ 76 = 2^2\\cdot 19\\textrm{ and }1216 = 2^6\\cdot 19. $ (This example was also found independently by Matthew Bolan, and by Dubickas, who posed it as part of the 2024 team selection test in Lithuania.) No other examples are known. This sequence is listed as A343101 at the OEIS.\nShorey and Tijdeman \\cite{ShTi16} have shown that, assuming a strong form of the ABC conjecture due to Baker, then the answer to the original problem is no.\nSee also [677].\nThe case of $x,y$ and $x+1,y+1$ appeared as Problem 1 in the Third Benelux Mathematical Olympiad 2011.\nThis problem is discussed in problem B19 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Ma68] Makowski, Andrzej, On a problem of {E}rd\\H{o}s. Enseign. Math. (2) (1968), 193.\n\n[ShTi16] Shorey, Tarlok N. and Tijdeman, Rob, Arithmetic properties of blocks of consecutive integers. (2016), 455--471.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2340, "problem_number": "EP-851", "title": "Erdős Problem #851", "statement": "Let $\\epsilon>0$. Is there some $r\\ll_\\epsilon 1$ such that the density of integers of the form $2^k+n$, where $k\\geq 0$ and $n$ has at most $r$ prime divisors, is at least $1-\\epsilon$?", "background": "Romanoff \\cite{Ro34} proved that the set of integers of the form $2^k+p$ (where $p$ is prime) has positive lower density.\nSee also [205].\nReferences\n\n\n[Ro34] Romanoff, N. P., \"{U}ber einige S\"Atze der additiven Zahlentheorie. Math. Ann. (1934), 668-678.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2341, "problem_number": "EP-852", "title": "Erdős Problem #852", "statement": "Let $d_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime. Let $h(x)$ be maximal such that for some $n(\\log x)^c $ for some constant $c>0$, and $ h(x)=o(\\log x)? $ ", "background": "Brun's sieve implies $h(x) \\to \\infty$ as $x\\to \\infty$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2342, "problem_number": "EP-853", "title": "Erdős Problem #853", "statement": "Let $d_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime. Let $r(x)$ be the smallest even integer $t$ such that $d_n=t$ has no solutions for $n\\leq x$.\nIs it true that $r(x)\\to \\infty$? Or even $r(x)/\\log x \\to \\infty$?", "background": "In \\cite{Er85c} Erdos omits the condition that $t$ be even, but this is clearly necessary.\nReferences\n\n\n[Er85c] Erdos, P., On some of my problems in number theory I would most like to see solved. Number theory (Ootacamund, 1984) (1985), 74-84.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2343, "problem_number": "EP-854", "title": "Erdős Problem #854", "statement": "Let $n_k$ denote the $k$th primorial, i.e. the product of the first $k$ primes.\nIf $1=a_1a$. Estimate the maximum of $ \\frac{1}{\\log N}\\sum_{n\\in A}\\frac{1}{n}. $ ", "background": "Alexander \\cite{Al66} and Erdos, S\\'{a}rk\"{o}zi, and Szemer\\'{e}di \\cite{ESS68} proved that this maximum is $o(1)$ (as $N\\to \\infty$). This condition on $A$ is a weaker form of the usual primitive condition. If $A$ is primitive then Behrend \\cite{Be35} proved $ \\frac{1}{\\log N}\\sum_{n\\in A}\\frac{1}{n}\\ll \\frac{1}{\\sqrt{\\log\\log N}}. $ An example of such a set $A$ is the set of all integers in $[N^{1/2},N]$ divisible by some prime $>N^{1/2}$.\nSee also [143].\nReferences\n\n\n[Al66] Alexander, Ralph, Density and multiplicative structure of sets of integers. Acta Arith. (1966/67), 321--332.\n\n[Be35] Behrend, F., On sequences of numbers not divisible by another. London Math. Soc. Journal (1935), 42-45.\n\n[ESS68] Erdos, P. and S\\'{a}rk\"ozi, A. and Szemer\\'{e}di, E., On the solvability of certain equations in sequences of\npositive upper logarithmic density. J. London Math. Soc. (1968), 71--78.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2347, "problem_number": "EP-859", "title": "Erdős Problem #859", "statement": "Let $t\\geq 1$ and let $d_t$ be the density of the set of integers $n\\in\\mathbb{N}$ for which $t$ can be represented as the sum of distinct divisors of $n$.\nDo there exist constants $c_1,c_2>0$ such that $ d_t \\sim \\frac{c_1}{(\\log t)^{c_2}} $ as $t\\to \\infty$?", "background": "Erdos \\cite{Er70} proved that $d_t$ always exists, and that there exist some constants $c_3,c_4>0$ such that $ \\frac{1}{(\\log t)^{c_3}} < d_t < \\frac{1}{(\\log t)^{c_4}}. $ \nReferences\n\n\n[Er70] Erdos, Paul, Some extremal problems in combinatorial number theory. Mathematical Essays Dedicated to A. J. Macintyre (1970), 123-133.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2348, "problem_number": "EP-860", "title": "Erdős Problem #860", "statement": "Let $h(n)$ be such that, for any $m\\geq 1$, in the interval $(m,m+h(n))$ there exist distinct integers $a_i$ for $1\\leq i\\leq \\pi(n)$ such that $p_i\\mid a_i$, where $p_i$ denotes the $i$th prime.\nEstimate $h(n)$.", "background": "A problem of Erdos and Pomerance \\cite{ErPo80}, who proved that $ h(n) \\ll \\frac{n^{3/2}}{(\\log n)^{1/2}}. $ Erdos and Selfridge proved $h(n)>(3-o(1))n$, and Ruzsa proved $h(n)/n\\to \\infty$.\nThis is discussed in problem B32 of Guy's collection \\cite{Gu04}.\nSee also [375].\nReferences\n\n\n[ErPo80] P. Erdos and C. Pomerance, Matching the natural numbers up to $n$ with distinct multiples of another interval. Indigationes Math. (1980), 147-151.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2349, "problem_number": "EP-863", "title": "Erdős Problem #863", "statement": "Let $r\\geq 2$ and let $A\\subseteq \\{1,\\ldots,N\\}$ be a set of maximal size such that there are at most $r$ solutions to $n=a+b$ with $a\\leq b$ for any $n$. (That is, $A$ is a $B_2[r]$ set.)\nSimilarly, let $B\\subseteq \\{1,\\ldots,N\\}$ be a set of maximal size such that there are at most $r$ solutions to $n=a-b$ for any $n$.\nIf $\\lvert A\\rvert\\sim c_rN^{1/2}$ as $N\\to \\infty$ and $\\lvert B\\rvert \\sim c_r'N^{1/2}$ as $N\\to \\infty$ then is it true that $c_r\neq c_r'$ for $r\\geq 2$? Is it true that $c_r'0$ such that, for all large $N$, if $A\\subseteq \\{1,\\ldots,N\\}$ has size at least $\\frac{5}{8}N+C$ then there are distinct $a,b,c\\in A$ such that $a+b,a+c,b+c\\in A$.", "background": "A problem of Erdos and S\\'{o}s (also earlier considered by Choi, Erdos, and Szemer\\'{e}di \\cite{CES75}, but Erdos had forgotten this). Taking all integers in $[N/8,N/4]$ and $[N/2,N]$ shows that $\\frac{5}{8}$ would be best possible here.\nIt is a classical folklore fact that if $A\\subseteq \\{1,\\ldots,2N\\}$ has size $\\geq N+2$ then there are distinct $a,b\\in A$ such that $a+b\\in A$, which establishes the $k=2$ case.\nIn general, one can define $f_k(N)$ to be minimal such that if $A\\subseteq \\{1,\\ldots,N\\}$ has size at least $f_k(N)$ then there are $k$ distinct $a_i\\in A$ such that all $\\binom{k}{2}$ pairwise sums are elements of $A$. Erdos and S\\'{o}s conjectured that $ f_k(N)\\sim \\frac{1}{2}\\left(1+\\sum_{1\\leq r\\leq k-2}\\frac{1}{4^r}\\right) N, $ and a similar example shows that this would be best possible.\nChoi, Erdos, and Szemer\\'{e}di \\cite{CES75} have proved that, for all $k\\geq 3$, there exists $\\epsilon_k>0$ such that (for large enough $N$) $ f_k(N)\\leq \\left(\\frac{2}{3}-\\epsilon_k\\right)N. $ \nReferences\n\n\n[CES75] Choi, S. L. G. and Erdos, P. and Szemer\\'{e}di, E., Some additive and multiplicative problems in number theory. Acta Arith. (1975), 37--50.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2352, "problem_number": "EP-866", "title": "Erdős Problem #866", "statement": "Let $k\\geq 3$ and $g_k(N)$ be minimal such that if $A\\subseteq \\{1,\\ldots,2N\\}$ has $\\lvert A\\rvert \\geq N+g_k(N)$ then there exist integers $b_1,\\ldots,b_k$ such that all $\\binom{k}{2}$ pairwise sums are in $A$ (but the $b_i$ themselves need not be in $A$).\nEstimate $g_k(N)$.", "background": "A problem of Choi, Erdos, and Szemer\\'{e}di. It is clear that, for the set of odd numbers in $\\{1,\\ldots,2N\\}$, no such $b_i$ exist, whence $g_k(N)\\geq 0$ always. Choi, Erdos, and Szemer\\'{e}di proved that $g_3(N)=2$ and $g_4(N) \\ll 1$. van Doorn has shown that $g_4(N)\\leq 2032$.\nChoi, Erdos, and Szemer\\'{e}di also proved that $ g_5(N)\\asymp \\log N $ and $ g_6(N)\\asymp N^{1/2}. $ In general they proved that $ g_k(N) \\ll_k N^{1-2^{-k}} $ and for every $\\epsilon>0$ if $k$ is sufficiently large then $ g_k(N) > N^{1-\\epsilon}. $ As an example, taking $A$ to be the set of all odd integers and the powers of $2$ shows that $g_5(N)\\gg \\log N$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2353, "problem_number": "EP-869", "title": "Erdős Problem #869", "statement": "If $A_1,A_2$ are disjoint additive bases of order $2$ (i.e. $A_i+A_i$ contains all large integers) then must $A=A_1\\cup A_2$ contain a minimal additive basis of order $2$ (one such that deleting any element creates infinitely many $n\not\\in A+A$)?", "background": "A question of Erdos and Nathanson \\cite{ErNa88}.\nHärtter \\cite{Ha56} and Nathanson \\cite{Na74} proved that there exist additive bases which do not contain any minimal additive bases.\nReferences\n\n\n[ErNa88] Erdos, Paul and Nathanson, Melvyn B., Partitions of bases into disjoint unions of bases. J. Number Theory (1988), 1--9.\n\n[Ha56] H\"{a}rtter, Erich, Ein Beitrag zur {T}heorie der {M}inimalbasen. J. Reine Angew. Math. (1956), 170--204.\n\n[Na74] Nathanson, Melvyn B., Minimal bases and maximal nonbases in additive number theory. J. Number Theory (1974), 324--333.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2354, "problem_number": "EP-870", "title": "Erdős Problem #870", "statement": "Let $k\\geq 3$ and $A$ be an additive basis of order $k$. Does there exist a constant $c=c(k)>0$ such that if $r(n)\\geq c\\log n$ for all large $n$ then $A$ must contain a minimal basis of order $k$? (Here $r(n)$ counts the number of representations of $n$ as the sum of at most $k$ elements from $A$.)", "background": "A question of Erdos and Nathanson \\cite{ErNa79}, who proved that this is true for $k=2$ if $1_A\\ast 1_A(n) > (\\log \\frac{4}{3})^{-1}\\log n$ for all large $n$.\nHärtter \\cite{Ha56} and Nathanson \\cite{Na74} proved that there exist additive bases which do not contain any minimal additive bases.\nSee also [868].\nReferences\n\n\n[ErNa79] Erdos, Paul and Nathanson, Melvyn B., Systems of distinct representatives and minimal bases in\nadditive number theory. (1979), 89--107.\n\n[Ha56] H\"{a}rtter, Erich, Ein Beitrag zur {T}heorie der {M}inimalbasen. J. Reine Angew. Math. (1956), 170--204.\n\n[Na74] Nathanson, Melvyn B., Minimal bases and maximal nonbases in additive number theory. J. Number Theory (1974), 324--333.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2355, "problem_number": "EP-872", "title": "Erdős Problem #872", "statement": "Consider the two-player game in which players alternately choose integers from $\\{2,3,\\ldots,n\\}$ to be included in some set $A$ (the same set for both players) such that no $a\\mid b$ for $a\neq b\\in A$.\nThe game ends when no legal move is possible. One player wants the game to last as long as possible, the other wants the game to end quickly. How long can the game be guaranteed to last for?\nAt least $\\epsilon n$ moves? (For $\\epsilon>0$ and $n$ sufficiently large.) At least $(1-\\epsilon)\\frac{n}{2}$ moves?", "background": "A number theoretic variant of a combinatorial game of Hajnal, in which players alternately add edges to a graph while keeping it triangle-free. This game must trivially end in at most $n^2/4$ moves, and F\"{u}redi and Seress \\cite{FuSe91} proved that it can be guaranteed to last for $\\gg n\\log n$ moves. Bir\\'{o}, Horn, and Wildstrom \\cite{BPW16} proved that it must end in at most $(\\frac{26}{121}+o(1))n^2$ moves.\nThis type of game is known as a saturation game.\nErdos does not specify which player goes first, which may result in different answers.\nReferences\n\n\n[BPW16] Bir\\'{o}, Csaba and Horn, Paul and Wildstrom, D. Jacob, An upper bound on the extremal version of Hajnal's\ntriangle-free game. Discrete Appl. Math. (2016), 20--28.\n\n[FuSe91] F\"{u}redi, Zolt\\'{a}n and Reimer, Dave and Seress, \\'{A}kos, Hajnal's triangle-free game and extremal graph problems. Congr. Numer. (1991), 123--128.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2356, "problem_number": "EP-873", "title": "Erdős Problem #873", "statement": "Let $A=\\{a_10$, there exists some $k$ such that $ F(A,X,k)H(n)-n^{1+o(1)}? $ Is it true that, for every $k\\geq 2$, if $n$ is sufficiently large then the admissible set which maximises $G(n)$ contains at least one integer with at least $k$ prime factors?", "background": "Erdos and Van Lint proved that $ H(n)-n^{3/2-o(1)}H(n)-n^{1+o(1)}$ assuming 'plausible (but hopeless) assumptions about the distribution of primes'. They also prove the second claim when $k=2$.\nSee also [878].\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2361, "problem_number": "EP-881", "title": "Erdős Problem #881", "statement": "Let $A\\subset\\mathbb{N}$ be an additive basis of order $k$ which is minimal, in the sense that if $B\\subset A$ is any infinite set then $A\\backslash B$ is not a basis of order $k$.\nMust there exist an infinite $B\\subset A$ such that $A\\backslash B$ is a basis of order $k+1$?\",\n \"difficulty\": \"L1\"\n},{", "background": "", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2362, "problem_number": "EP-883", "title": "Erdős Problem #883", "statement": "For $A\\subseteq \\{1,\\ldots,n\\}$ let $G(A)$ be the graph with vertex set $A$, where two integers are joined by an edge if they are coprime.\nIs it true that if $ \\lvert A\\rvert >\\lfloor\\tfrac{n}{2}\\rfloor+\\lfloor\\tfrac{n}{3}\\rfloor-\\lfloor\\tfrac{n}{6}\\rfloor $ then $G(A)$ contains all odd cycles of length $\\leq \\frac{n}{3}+1$?\nIs it true that, for every $\\ell\\geq 1$, if $n$ is sufficiently large and $ \\lvert A\\rvert >\\lfloor\\tfrac{n}{2}\\rfloor+\\lfloor\\tfrac{n}{3}\\rfloor-\\lfloor\\tfrac{n}{6}\\rfloor $ then $G(A)$ must contain a complete $(1,\\ell,\\ell)$ triparite graph on $2\\ell+1$ vertices?", "background": "A problem of Erdos and S\\'{a}rk\\H{o}zy \\cite{ErSa97}, who prove that if $ \\lvert A\\rvert >\\lfloor\\tfrac{n}{2}\\rfloor+\\lfloor\\tfrac{n}{3}\\rfloor-\\lfloor\\tfrac{n}{6}\\rfloor $ then $G(A)$ contains all odd cycles of length $\\leq cn$ for some constant $c>0$.\nThis threshold is the best possible, since one could take $A$ to be the set of $m\\leq n$ which are divisible by either $2$ or $3$, in which case $G(A)$ contains no triangles.\nThe second question was solved by S\\'{a}rk\"{o}zy \\cite{Sa99} who proved that, for large $n$, if $\\lvert A\\rvert$ exceeds the given threshold then $G(A)$ contains a complete $(1,\\ell,\\ell)$ triparite graph with $ \\ell \\gg \\frac{\\log n}{\\log\\log n}. $ \nReferences\n\n\n[ErSa97] Erdos, Paul and Sarkozy, Gabor N., On cycles in the coprime graph of integers. Electron. J. Combin. (1997), Research Paper 8, approx. 11.\n\n[Sa99] S\\'ark\"ozy, G\\'abor N., Complete tripartite subgraphs in the coprime graph of\nintegers. Discrete Math. (1999), 227--238.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2363, "problem_number": "EP-884", "title": "Erdős Problem #884", "statement": "Is it true that, for any $n$, if $d_1<\\cdots 0$. Is it true that, for all large $n$, the number of divisors of $n$ in $(n^{1/2},n^{1/2}+n^{1/2-\\epsilon})$ is $O_\\epsilon(1)$?", "background": "Erdos attributes this conjecture to Ruzsa. Erdos and Rosenfeld \\cite{ErRo97} proved that there are infinitely many $n$ such that there are four divisors of $n$ in $(n^{1/2},n^{1/2}+n^{1/4})$.\nSee also [887].\nReferences\n\n\n[ErRo97] Erdos, Paul and Rosenfeld, Moshe, The factor-difference set of integers. Acta Arith. (1997), 353--359.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2366, "problem_number": "EP-887", "title": "Erdős Problem #887", "statement": "Is there an absolute constant $K$ such that, for every $C>0$, if $n$ is sufficiently large then $n$ has at most $K$ divisors in $(n^{1/2},n^{1/2}+C n^{1/4})$.", "background": "A question of Erdos and Rosenfeld \\cite{ErRo97}, who proved that there are infinitely many $n$ with $4$ divisors in $(n^{1/2},n^{1/2}+n^{1/4})$, and ask whether $4$ is best possible here.\nReferences\n\n\n[ErRo97] Erdos, Paul and Rosenfeld, Moshe, The factor-difference set of integers. Acta Arith. (1997), 353--359.\n\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2367, "problem_number": "EP-888", "title": "Erdős Problem #888", "statement": "What is the size of the largest $A\\subseteq \\{1,\\ldots,n\\}$ such that if $a\\leq b\\leq c\\leq d\\in A$ are such that $abcd$ is a square then $ad=bc$?", "background": "A question of Erdos, S\\'{a}rk\"{o}zy, and S\\'{o}s. Erdos claims that S\\'{a}rk\"{o}zy proved that $\\lvert A\\rvert =o(n)$ (a proof of this bound is provided by Tao in the comments).\nThe primes show that $\\lvert A\\rvert \\gg n/\\log n$ is possible. Cambie and Weisenberg have noted in the comments that the set of semiprimes also works, showing $ (1+o(1))\\frac{\\log\\log n}{\\log n}n \\leq \\lvert A\\rvert $ is achievable.\nSee also [121].\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2368, "problem_number": "EP-889", "title": "Erdős Problem #889", "statement": "For $k\\geq 0$ and $n\\geq 1$ let $v(n,k)$ count the prime factors of $n+k$ which do not divide $n+i$ for $0\\leq ik$.\nIs it true that $ v_0(n)=\\max_{k\\geq 0}v(n,k)\\to \\infty $ as $n\\to \\infty$?", "background": "A question of Erdos and Selfridge \\cite{ErSe67}, who could only show that $v_0(n)\\geq 2$ for $n\\geq 17$. More generally, they conjecture that $ v_l(n)=\\max_{k\\geq l}v(n,k)\\to \\infty $ as $n\\to \\infty$, for every fixed $l$, but could not even prove that $v_1(n)\\geq 2$ for all large $n$.\nThis is problem B27 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[ErSe67] Erdos, P. and Selfridge, J. L., Some problems on the prime factors of consecutive integers. Illinois J. Math. (1967), 428--430.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2369, "problem_number": "EP-890", "title": "Erdős Problem #890", "statement": "If $\\omega(n)$ counts the number of distinct prime factors of $n$, then is it true that, for every $k\\geq 1$, $ \\liminf_{n\\to \\infty}\\sum_{0\\leq ik$ by P\\'{o}lya's theorem.\nIt is a classical fact that $ \\limsup_{n\\to \\infty}\\omega(n)\\frac{\\log\\log n}{\\log n}=1. $ \nReferences\n\n\n[ErSe67] Erdos, P. and Selfridge, J. L., Some problems on the prime factors of consecutive integers. Illinois J. Math. (1967), 428--430.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2370, "problem_number": "EP-891", "title": "Erdős Problem #891", "statement": "Let $2=p_1k$ many prime factors?", "background": "Schinzel deduced from P\\'{o}lya's theorem \\cite{Po18} (that the sequence of $k$-smooth integers has unbounded gaps) that this is true with $p_1\\cdots p_k$ replaced by $p_1\\cdots p_{k-1}p_{k+1}$.\nThis is unknown even for $k=2$ - that is, is it true that in every interval of $6$ (sufficiently large) consecutive integers there must exist one with at least $3$ prime factors?\nWeisenberg has observed that Dickson's conjecture implies the answer is no if we replace $p_1\\cdots p_k$ with $p_1\\cdots p_k-1$. Indeed, let $L_k$ be the lowest common multiple of all integers at most $p_1\\cdots p_k$. By Dickson's conjecture there are infinitely many $n'$ such that $\\frac{L_k}{m}n'+1$ is prime for all $1\\leq m f(C) e? $ \",\n \"difficulty\": \"L1\"\n},{", "background": "", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2378, "problem_number": "EP-912", "title": "Erdős Problem #912", "statement": "If $ n! = \\prod_i p_i^{k_i} $ is the factorisation into distinct primes then let $h(n)$ count the number of distinct exponents $k_i$.\nProve that there exists some $c>0$ such that $ h(n) \\sim c \\left(\\frac{n}{\\log n}\\right)^{1/2} $ as $n\\to \\infty$.", "background": "A problem of Erdos and Selfridge, who proved (see \\cite{Er82c}) $ h(n) \\asymp \\left(\\frac{n}{\\log n}\\right)^{1/2}. $ A heuristic of Tao using the Cram\\'{e}r model for the primes (detailed in the comments) suggests this is true with $ c=\\sqrt{2\\pi}=2.506\\cdots. $ \nReferences\n\n\n[Er82c] Erdos, P., Miscellaneous problems in number theory. Congr. Numer. (1982), 25-45.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2379, "problem_number": "EP-913", "title": "Erdős Problem #913", "statement": "Are there infinitely many $n$ such that if $ n(n+1) = \\prod_i p_i^{k_i} $ is the factorisation into distinct primes then all exponents $k_i$ are distinct?", "background": "It is likely that there are infinitely many primes $p$ such that $8p^2-1$ is also prime, in which case this is true with exponents $\\{1,2,3\\}$, letting $n=8p^2-1$.\nThis problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2380, "problem_number": "EP-917", "title": "Erdős Problem #917", "statement": "Let $k\\geq 4$ and $f_k(n)$ be the largest number of edges in a graph on $n$ vertices which has chromatic number $k$ and is critical (i.e. deleting any edge reduces the chromatic number).\nIs it true that $ f_k(n) \\gg_k n^2? $ Is it true that $ f_6(n)\\sim n^2/4? $ More generally, is it true that, for $k\\geq 6$, $ f_k(n) \\sim \\frac{1}{2}\\left(1-\\frac{1}{\\lfloor k/3\\rfloor}\\right)n^2? $ ", "background": "Erdos \\cite{Er93} wrote 'I learned of this definition from Dirac in 1949 and immediately asked whether $f_k(n)=o(n^2)$. To my great surprise Dirac constructed a $6$ critical graph on $n$ vertices with more than $\\frac{n^2}{4}$ edges.' In fact Dirac \\cite{Di52} proved $ f_6(4n+2) \\geq 4n^2+8n+3, $ as witnessed by taking two disjoint copies of $C_{2n+1}$ and adding all edges between them.\nErdos \\cite{Er69b} observed that Dirac's construction generalises to show that, if $3\\mid k$, there are infinitely many values of $n$ (those of the shape $mk/3$ where $m$ is odd) such that $ f_k(n) \\geq \\frac{1}{2}\\left(1-\\frac{1}{k/3}\\right)n^2 + n. $ Toft \\cite{To70} proved that $f_k(n)\\gg_k n^2$ for $k\\geq 4$.\nConstructions of Stiebitz \\cite{St87} show that, for $k\\geq 6$, there exist infinitely many values of $n$ such that $ f_k(n) \\geq \\frac{1}{2}\\left(1-\\frac{1}{\\lfloor k/3\\rfloor+\\delta_k}\\right)n^2 $ where $\\delta_k=0$ if $k\\equiv 0\\pmod{3}$, $\\delta_k=1/7$ if $k\\equiv 1\\pmod{3}$, and $\\delta_k\\equiv 24/69$ if $k\\equiv 2\\pmod{3}$, which disproves Erdos' conjectured asympotic for $k\not\\equiv 0\\pmod{3}$.\nStiebitz also proved the general upper bound $ f_k(n) < \\mathrm{ex}(n;K_{k-1})\\sim \\frac{1}{2}\\left(1-\\frac{1}{k-2}\\right)n^2 $ for large $n$. Luo, Ma, and Yang \\cite{LMY23} have improved this upper bound to $ f_k(n) \\leq \\frac{1}{2}\\left(1-\\frac{1}{k-2}-\\frac{1}{36(k-1)^2}+o(1)\\right)n^2 $ See also [944] and [1032].\nReferences\n\n\n[Di52] Dirac, G. A., A property of {$4$}-chromatic graphs and some remarks on\ncritical graphs. J. London Math. Soc. (1952), 85-92.\n\n[Er69b] Erdos, P., Problems and results in chromatic graph theory. Proof Techniques in Graph Theory (Proc. Second Ann\nArbor Graph Theory Conf., Ann Arbor, Mich.,\n1968) (1969), 27-35.\n\n[Er93] Erdos, Paul, Some of my favorite solved and unsolved problems in graph\ntheory. Quaestiones Math. (1993), 333-350.\n\n[LMY23] Luo, Cong and Ma, Jie and Yang, Tianchi, On the maximum number of edges in {$k$}-critical graphs. Combin. Probab. Comput. (2023), 900--911.\n\n[St87] Stiebitz, M., Subgraphs of colour-critical graphs. Combinatorica (1987), 303--312.\n\n[To70] Toft, B., On the maximal number of edges of critical {$k$}-chromatic\ngraphs. Studia Sci. Math. Hungar. (1970), 461--470.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2381, "problem_number": "EP-918", "title": "Erdős Problem #918", "statement": "Is there a graph with $\\aleph_2$ vertices and chromatic number $\\aleph_2$ such that every subgraph on $\\aleph_1$ vertices has chromatic number $\\leq\\aleph_0$?\nIs there a graph with $\\aleph_{\\omega+1}$ vertices and chromatic number $\\aleph_1$ such that every subgraph on $\\aleph_\\omega$ vertices has chromatic number $\\leq\\aleph_0$?", "background": "A question of Erdos and Hajnal \\cite{ErHa68b}, who proved that for every finite $k$ there is a graph with chromatic number $\\aleph_1$ where each subgraph on less than $\\aleph_k$ vertices has chromatic number $\\leq \\aleph_0$.\nIn \\cite{Er69b} it is asked with chromatic number $=\\aleph_0$, but in the comments louisd observes this is (assuming subgraph and not induced subgraph was intended) trivially impossible, and hence presumably the problem was intended as written here (which is how it is posed in \\cite{ErHa68b}).\nReferences\n\n\n[Er69b] Erdos, P., Problems and results in chromatic graph theory. Proof Techniques in Graph Theory (Proc. Second Ann\nArbor Graph Theory Conf., Ann Arbor, Mich.,\n1968) (1969), 27-35.\n\n[ErHa68b] Erdos, P. and Hajnal, A., On chromatic number of infinite graphs. (1968), 83--98.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2382, "problem_number": "EP-919", "title": "Erdős Problem #919", "statement": "Is there a graph $G$ with vertex set $\\omega_2^2$ and chromatic number $\\aleph_2$ such that every subgraph whose vertices have a lesser type has chromatic number $\\leq \\aleph_0$?\nWhat if instead we ask for $G$ to have chromatic number $\\aleph_1$?", "background": "This question was inspired by a theorem of Babai, that if $G$ is a graph on a well-ordered set with chromatic number $\\geq \\aleph_0$ there is a subgraph on vertices with order-type $\\omega$ with chromatic number $\\aleph_0$.\nErdos and Hajnal showed this does not generalise to higher cardinals - they (see \\cite{Er69b}) constructed a set on $\\omega_1^2$ with chromatic number $\\aleph_1$ such that every strictly smaller subgraph has chromatic number $\\leq \\aleph_0$ as follows: the vertices of $G$ are the pairs $(x_\\alpha,y_\\beta)$ for $1\\leq \\alpha,\\beta <\\omega_1$, ordered lexicographically. We connect $(x_{\\alpha_1},y_{\\beta_1})$ and $(x_{\\alpha_2},y_{\\beta_2})$ if and only if $\\alpha_1<\\alpha_2$ and $\\beta_1<\\beta_2$.\nA similar construction produces a graph on $\\omega_2^2$ with chromatic number $\\aleph_2$ such that every smaller subgraph has chromatic number $\\leq \\aleph_1$.\nReferences\n\n\n[Er69b] Erdos, P., Problems and results in chromatic graph theory. Proof Techniques in Graph Theory (Proc. Second Ann\nArbor Graph Theory Conf., Ann Arbor, Mich.,\n1968) (1969), 27-35.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2383, "problem_number": "EP-920", "title": "Erdős Problem #920", "statement": "Let $f_k(n)$ be the maximum possible chromatic number of a graph with $n$ vertices which contains no $K_k$.\nIs it true that, for $k\\geq 4$, $ f_k(n) \\gg \\frac{n^{1-\\frac{1}{k-1}}}{(\\log n)^{c_k}} $ for some constant $c_k>0$?", "background": "Graver and Yackel \\cite{GrYa68} proved that $ f_k(n) \\ll \\left(n\\frac{\\log\\log n}{\\log n}\\right)^{1-\\frac{1}{k-1}}. $ It is known that $f_3(n)\\asymp (n/\\log n)^{1/2}$ (see [1104]).\nThe lower bound $R(4,m) \\gg m^3/(\\log m)^4$ of Mattheus and Verstraete \\cite{MaVe23} (see [166]) implies $ f_4(n) \\gg \\frac{n^{2/3}}{(\\log n)^{4/3}}. $ A positive answer to this question would follow from [986]. The known bounds for that problem imply $ f_k(n) \\gg \\frac{n^{1-\\frac{2}{k+1}}}{(\\log n)^{c_k}}. $ See [1104] (and also [1013]) for the case $k=3$.\nReferences\n\n\n[GrYa68] Graver, Jack E. and Yackel, James, Some graph theoretic results associated with Ramsey's theorem. J. Combinatorial Theory (1968), 125--175.\n\n[MaVe23] Mattheus, S. and Verstraete, J., The asymptotics of $r(4,t)$. arXiv:2306.04007 (2023).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2384, "problem_number": "EP-928", "title": "Erdős Problem #928", "statement": "Let $\\alpha,\\beta\\in (0,1)$ and let $P(n)$ denote the largest prime divisor of $n$. Does the density of integers $n$ such that $P(n) k^{1/2-o(1)}$.\nIt is trivial that $S(k)\\leq k+1$ since, for example, one can take $n\\equiv 1\\pmod{(k+1)!}$. The best bound on large gaps between primes due to Ford, Green, Konyagin, Maynard, and Tao \\cite{FGKMT18} (see [4]) implies $ S(k) \\ll k \\frac{\\log\\log\\log k}{\\log\\log k\\log\\log\\log\\log k}. $ \nReferences\n\n\n[FGKMT18] Ford, Kevin and Green, Ben and Konyagin, Sergei and Maynard, James and Tao, Terence, Long gaps between primes. J. Amer. Math. Soc. (2018), 65-105.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2386, "problem_number": "EP-930", "title": "Erdős Problem #930", "statement": "Is it true that, for every $r$, there is a $k$ such that if $I_1,\\ldots,I_r$ are disjoint intervals of consecutive integers, all of length at least $k$, then $ \\prod_{1\\leq i\\leq r}\\prod_{m\\in I_i}m $ is not a perfect power?", "background": "Erdos and Selfridge \\cite{ErSe75} proved that the product of consecutive integers is never a power (establishing the case $r=1$). The condition that the intervals be large in terms of $r$ is necessary for $r=2$ - see the constructions in [363].\nSee also [363] for the case of squares.\nReferences\n\n\n[ErSe75] Erdos, P. and Selfridge, J. L., The product of consecutive integers is never a power. Illinois J. Math. (1975), 292-301.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2387, "problem_number": "EP-931", "title": "Erdős Problem #931", "statement": "Let $k_1\\geq k_2\\geq 3$. Are there only finitely many $n_2\\geq n_1+k_1$ such that $ \\prod_{1\\leq i\\leq k_1}(n_1+i)\\textrm{ and }\\prod_{1\\leq j\\leq k_2}(n_2+j) $ have the same prime factors?", "background": "Tijdeman gave the example $ 19,20,21,22\\textrm{ and }54,55,56,57. $ Erdos \\cite{Er76d} was unsure of this conjecture, and thought perhaps if the two products have the same prime factors then $n_2>2(n_1+k_1)$. It is not clear but it is possible that he meant to ask this question also permitting finitely many counterexamples. Indeed, without this caveat it is false - AlphaProof has found the counterexample $ 10! = 2^8\\cdot 3^4\\cdot 5^2\\cdot 7 $ and $ 14\\cdot 15\\cdot 16 = 2^5\\cdot 3\\cdot 5\\cdot 7, $ so that $n_1=0$, $k_1=10$, $n_2=13$, and $k_2=3$.\nSee also [388].\nThis is discussed in problem B35 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Er76d] Erdos, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2388, "problem_number": "EP-932", "title": "Erdős Problem #932", "statement": "Let $p_k$ denote the $k$th prime. For infinitely many $r$ there are at least two integers $p_rn\\log n. $ Steinerberger has noted a simple proof of this fact follows from taking $n=2^{3^r}$ for any integer $r\\geq 1$, when $k=3^r$ and $l=r+1$.\nReferences\n\n\n[Er76d] Erdos, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2390, "problem_number": "EP-934", "title": "Erdős Problem #934", "statement": "Let $h_t(d)$ be minimal such that every graph $G$ with $h_t(d)$ edges and maximal degree $\\leq d$ contains two edges whose shortest path between them has length $\\geq t$.\nEstimate $h_t(d)$.", "background": "A problem of Erdos and Ne\\v{s}et\\v{r}il. Erdos \\cite{Er88} wrote 'This problem seems to be interesting only if there is a nice expression for $h_t(d)$.'\nIt is easy to see that $h_t(d)\\leq 2d^t$ always and $h_1(d)=d+1$.\nErdos and Ne\\v{s}et\\v{r}il and Bermond, Bond, Paoli, and Peyrat \\cite{BBPP83} independently conjectured that $h_2(d) \\leq \\tfrac{5}{4}d^2+1$, with equality for even $d$ (see [149]). This was proved by Chung, Gy\\'{a}rf\\'{a}s, Tuza, and Trotter \\cite{CGTT90}.\nCambie, Cames van Batenburg, de Joannis de Verclos, and Kang \\cite{CCJK22} conjectured that $ h_3(d) \\leq d^3-d^2+d+2, $ with equality if and only if $d=p^k+1$ for some prime power $p^k$, and proved that $h_3(3)=23$. They also conjecture that, for all $t\\geq 3$, $h_t(d)\\geq (1-o(1))d^t$ for infinitely many $d$ and $h_t(d)\\leq (1+o(1))d^t$ for all $d$ (where the $o(1)$ term $\\to 0$ as $d\\to \\infty$).\nThe same authors prove that, if $t$ is large, then there are infinitely many $d$ such that $h_t(d) \\geq 0.629^td^t$, and that for all $t\\geq 1$ we have $ h_t(d) \\leq \\tfrac{3}{2}d^t+1. $ \nReferences\n\n\n[BBPP83] Bermond, J.-C. and Bond, J. and Paoli, M. and Peyrat, C., Graphs and interconnection networks: diameter and\nvulnerability. (1983), 1--30.\n\n[CCJK22] Cambie, Stijn and Cames van Batenburg, Wouter and de Joannis\nde Verclos, R\\'{e}mi and Kang, Ross J., Maximizing line subgraphs of diameter at most {$t$}. SIAM J. Discrete Math. (2022), 939--950.\n\n[CGTT90] Chung, F. R. K. and Gy\\'arf\\'as, A. and Tuza, Z. and Trotter,\nW. T., The maximum number of edges in {$2K_2$}-free graphs of bounded\ndegree. Discrete Math. (1990), 129--135.\n\n[Er88] Erdos, P, Problems and results in combinatorial analysis and graph theory. Discrete Math. (1988), 81-92.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2391, "problem_number": "EP-935", "title": "Erdős Problem #935", "statement": "For any integer $n=\\prod p^{k_p}$ let $Q_2(n)$ be the powerful part of $n$, so that $ Q_2(n) = \\prod_{\\substack{p\\\\ k_p\\geq 2}}p^{k_p}. $ Is it true that, for every $\\epsilon>0$ and $\\ell\\geq 1$, if $n$ is sufficiently large then $ Q_2(n(n+1)\\cdots(n+\\ell))2$, only keeping those prime powers with exponent $\\geq r$.\nReferences\n\n\n[Er76d] Erdos, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2392, "problem_number": "EP-936", "title": "Erdős Problem #936", "statement": "Are $ 2^n\\pm 1 $ and $ n!\\pm 1 $ powerful (i.e. if $p\\mid m$ then $p^2\\mid m$) for only finitely many $n$?", "background": "Cushing and Pascoe \\cite{CuPa16} have shown the answer to the second question is yes assuming the abc conjecture - in fact, for any fixed $k\\geq 0$, there are only finitely many $n$ and powerful $x$ such that $\\lvert x-n!\\rvert \\leq k$.\nCrowdMath \\cite{Cr20} has shown that the answer to the first question is yes, again assuming the abc conjecture.\nReferences\n\n\n[Cr20] P. A. CrowdMath, Applications of the abc conjecture to powerful numbers. arXiv:2005.07321 (2020).\n\n[CuPa16] D. Cushing and J. E. Pascoe, Powerful numbers and the ABC-conjecture. arXiv:1611.01192 (2016).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2393, "problem_number": "EP-938", "title": "Erdős Problem #938", "statement": "Let $A=\\{n_10$ such that $ h(n) < (\\log n)^{c+o(1)} $ and, for infinitely many $n$, $ h(n) >(\\log n)^{c-o(1)}? $ ", "background": "Erdos writes it is not hard to prove that $\\limsup h(n)=\\infty$, and that the density $\\delta_l$ of integers for which $h(n)=l$ exists and $\\sum \\delta_l=1$.\nA proof that $h(n)$ is unbounded is provided by van Doorn in the comments.\nDe Koninck and Luca \\cite{DeLu04} have proved, for infinitely many $n$, $ h(n) \\gg \\left(\\frac{\\log n}{\\log\\log n}\\right)^{1/3}. $ They also give the density ($\\approx 0.275$) of those $n$ such that $h(n)=1$.\nReferences\n\n\n[DeLu04] De Koninck, Jean-Marie and Luca, Florian, Sur la proximit\\'{e} des nombres puissants. Acta Arith. (2004), 149--157.\n\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2397, "problem_number": "EP-943", "title": "Erdős Problem #943", "statement": "Let $A$ be the set of powerful numbers (if $p\\mid n$ then $p^2\\mid n$). Is it true that $ 1_A\\ast 1_A(n)=n^{o(1)} $ for every $n$?\",\n \"difficulty\": \"L1\"\n},{", "background": "", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2398, "problem_number": "EP-944", "title": "Erdős Problem #944", "statement": "A critical vertex, edge, or set of edges, is one whose deletion lowers the chromatic number.\nLet $k\\geq 4$ and $r\\geq 1$. Must there exist a graph $G$ with chromatic number $k$ such that every vertex is critical, yet every critical set of edges has size $>r$?", "background": "A graph $G$ with chromatic number $k$ in which every vertex is critical is called $k$-vertex-critical.\nThis was conjectured by Dirac in 1970 for $k\\geq 4$ and $r=1$. Dirac's conjecture was proved, for $k=5$, by Brown \\cite{Br92}. Lattanzio \\cite{La02} proved there exist such graphs for all $k$ such that $k-1$ is not prime. Independently, Jensen \\cite{Je02} gave an alternative construction for all $k\\geq 5$. The case $k=4$ and $r=1$ remains open.\nMartinsson and Steiner \\cite{MaSt25} proved this is true for every $r\\geq 1$ if $k$ is sufficiently large, depending on $r$. Skottova and Steiner \\cite{SkSt25} have improved this, proving that such graphs exist for all $k\\geq 5$ and $r\\geq 1$. The only remaining open case is $k=4$ (even the case $k=4$ and $r=1$ remains open).\nErdos also asked a stronger quantitative form of this question: let $f_k(n)$ denote the largest $r\\geq 1$ such that there exists a $k$-vertex-critical graph on $n$ vertices such that no set of at most $r$ edges is critical. He then asks whether $f_k(n)\\to \\infty$ as $n\\to \\infty$. Skottova and Steiner \\cite{SkSt25} have proved this for $k\\geq 5$, establishing the bounds $ n^{1/3}\\ll_k f_k(n) \\ll_k \\frac{n}{(\\log n)^C} $ for all $k\\geq 5$, where $C>0$ is an absolute constant.\nThis is Problem 91 in the graph problems collection. See also [917] and [1032].\nReferences\n\n\n[Br92] Brown, Jason I., A vertex critical graph without critical edges. Discrete Math. (1992), 99--101.\n\n[Je02] Jensen, Tommy R., Dense critical and vertex-critical graphs. Discrete Math. (2002), 63--84.\n\n[La02] Lattanzio, John J., A note on a conjecture of {D}irac. Discrete Math. (2002), 323--330.\n\n[MaSt25] Martinsson, Anders and Steiner, Raphael, Vertex-critical graphs far from edge-criticality. Combin. Probab. Comput. (2025), 151--157.\n\n[SkSt25] E. Skottova and R. Steiner, Critical edge sets in vertex-critical graphs. arXiv:2508.08703 (2025).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2399, "problem_number": "EP-945", "title": "Erdős Problem #945", "statement": "Let $F(x)$ be the maximal $k$ such that there exist $n+1,\\ldots,n+k\\leq x$ with $\\tau(n+1),\\ldots,\\tau(n+k)$ all distinct (where $\\tau(m)$ counts the divisors of $m$). Estimate $F(x)$. In particular, is it true that $ F(x) \\leq (\\log x)^{O(1)}? $ In other words, is there a constant $C>0$ such that, for all large $x$, every interval $[x,x+(\\log x)^C]$ contains two integers with the same number of divisors?", "background": "A problem of Erdos and Mirsky \\cite{ErMi52}, who proved that $ \\frac{(\\log x)^{1/2}}{\\log\\log x}\\ll F(x) \\ll \\exp\\left(O\\left(\\frac{(\\log x)^{1/2}}{\\log\\log x}\\right)\\right). $ Erdos \\cite{Er85e} claimed that the lower bound could be improved via their method 'with some more work' to $(\\log x)^{1-o(1)}$. Beker has improved the upper bound to $ F(x) \\ll \\exp\\left(O\\left((\\log x)^{1/3+o(1)}\\right)\\right). $ Cambie has observed that Cram\\'{er's conjecture} implies that $F(x) \\ll (\\log x)^2$, and furthermore if every interval in $[x,2x]$ of length $\\gg \\log x$ contains a squarefree number (see [208]) then every interval of length $\\gg (\\log x)^2$ contains two numbers with the same number of divisors, whence $ F(x) \\ll (\\log x)^2. $ See [1004] for the analogous problem with the Euler totient function.\nThis problem is discussed in problem B18 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Er85e] Erdos, P., Some problems and results in number theory. Number theory and combinatorics. Japan 1984 (Tokyo,\nOkayama and Kyoto, 1984) (1985), 65-87.\n\n[ErMi52] Erdos, P. and Mirsky, L., The distribution of values of the divisor function {$d(n)$}. Proc. London Math. Soc. (3) (1952), 257--271.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2400, "problem_number": "EP-948", "title": "Erdős Problem #948", "statement": "Is there a function $f(n)$ and a $k$ such that in any $k$-colouring of the integers there exists a sequence $a_1<\\cdots$ such that $a_n0$ such that there are $\\gg n^c/\\log n$ many primes in $[n,n+n^c]$ implies that $\\liminf f(n)>0$.\nErdos writes that a 'weaker conjecture which is perhaps not quite inaccessible' is that, for every $\\epsilon>0$, if $x$ is sufficiently large there exists $y0$ then $f(n)\\ll \\log\\log\\log n$.\nThe study of $f(p)$ is even harder, and Erdos could not prove that $ \\sum_{p0$ such that $h(n)>n^{1+c}$ for all large $n$.", "background": "A problem of Erdos and Pach \\cite{ErPa90}, who proved that $h(n) \\ll n^{4/3}$. They also consider the related function where we consider $n$ disjoint convex sets (not necessarily translates), for which they give an upper bound of $\\ll n^{7/5}$.\nIt is trivial that $h(n)\\geq f(n)$, where $f(n)$ is the maximal number of unit distances determined by $n$ points in $\\mathbb{R}^2$ (see [90]).\nReferences\n\n\n[ErPa90] Erdos, P. and Pach, J., Variations on the theme of repeated distances. Combinatorica (1990), 261--269.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2409, "problem_number": "EP-959", "title": "Erdős Problem #959", "statement": "Let $A\\subset \\mathbb{R}^2$ be a set of size $n$ and let $\\{d_1,\\ldots,d_k\\}$ be the set of distinct distances determined by $A$. Let $f(d)$ be the number of times the distance $d$ is determined, and suppose the $d_i$ are ordered such that $ f(d_1)\\geq f(d_2)\\geq \\cdots \\geq f(d_k). $ Estimate $ \\max (f(d_1)-f(d_2)), $ where the maximum is taken over all $A$ of size $n$.", "background": "More generally, one can ask about $ \\max (f(d_r)-f(d_{r+1})). $ Clemen, Dumitrescu, and Liu \\cite{CDL25}, have shown that $ \\max (f(d_1)-f(d_2))\\gg n\\log n. $ More generally, for any $1\\leq k\\leq \\log n$, there exists a set $A$ of $n$ points such that $ f(d_r)-f(d_{r+1})\\gg \\frac{n\\log n}{r}. $ They conjecture that $n\\log n$ can be improved to $n^{1+c/\\log\\log n}$ for some constant $c>0$.\nReferences\n\n\n[CDL25] F. Clemen, A. Dumitrescu, and D. Liu, On multiplicities of interpoint distances. arXiv:2505.04283 (2025).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2410, "problem_number": "EP-960", "title": "Erdős Problem #960", "statement": "Let $r,k\\geq 2$ be fixed. Let $A\\subset \\mathbb{R}^2$ be a set of $n$ points with no $k$ points on a line. Determine the threshold $f_{r,k}(n)$ such that if there are at least $f_{r,k}(n)$ many ordinary lines (lines containing exactly two points) then there is a set $A'\\subseteq A$ of $r$ points such that all $\\binom{r}{2}$ many lines determined by $A'$ are ordinary.\nIs it true that $f_{r,k}(n)=o(n^2)$, or perhaps even $\\ll n$?", "background": "Tur\\'{a}n's theorem implies $ f_{r,k}(n) \\leq \\left(1-\\frac{1}{r-1}\\right)\\frac{n^2}{2}+1. $ See also [209].\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2411, "problem_number": "EP-961", "title": "Erdős Problem #961", "statement": "Let $f(k)$ be the minimal $n$ such that every set of $n$ consecutive integers $>k$ contains an integer divisible by a prime $>k$. Estimate $f(k)$.", "background": "In other words, how large can a consecutive set of $k$-smooth integers be? Sylvester and Schur (see \\cite{Er34}) proved $f(k)\\leq k$ and Erdos \\cite{Er55d} proved $ f(k)<3\\frac{k}{\\log k}. $ Jutila \\cite{Ju74} and Ramachandra, and Shorey \\cite{RaSh73} proved $ f(k) \\ll \\frac{\\log\\log\\log k}{\\log \\log k}\\frac{k}{\\log k}. $ It is likely that $f(k) \\ll (\\log k)^{O(1)}$.\nThis is essentially equivalent to [683].\nReferences\n\n\n[Er34] Erdos, Paul, A {T}heorem of {S}ylvester and {S}chur. J. London Math. Soc. (1934), 282--288.\n\n[Er55d] Erdos, P., On consecutive integers. Nieuw Arch. Wisk. (3) (1955), 124--128.\n\n[Ju74] Jutila, Matti, On numbers with a large prime factor. {II}. J. Indian Math. Soc. (N.S.) (1974), 125--130.\n\n[RaSh73] Ramachandra, K. and Shorey, T. N., On gaps between numbers with a large prime factor. Acta Arith. (1973), 99--111.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2412, "problem_number": "EP-962", "title": "Erdős Problem #962", "statement": "Let $k(n)$ be the maximal $k$ such that there exists $m\\leq n$ such that each of the integers $ m+1,\\ldots,m+k $ are divisible by at least one prime $>k$. Estimate $k(n)$.", "background": "Erdos \\cite{Er65} wrote it is 'not hard to prove' that $ k(n)\\gg_\\epsilon \\exp((\\log n)^{1/2-\\epsilon}) $ and it 'seems likely' that $k(n)=o(n^\\epsilon)$, but had no non-trivial upper bound for $k(n)$.\nIt is not clear what he meant by a non-trivial bound for this problem, but Tao in the comments has given a simple argument proving $k(n) \\leq (1+o(1))n^{1/2}$.\nTang has proved a lower bound of $ k(n)\\geq \\exp\\left(\\left(\\frac{1}{\\sqrt{2}}-o(1)\\right)\\sqrt{\\log n\\log\\log n}\\right). $ \nReferences\n\n\n[Er65] Erdos, P., Extremal problems in number theory. Proc. Sympos. Pure Math., Vol. VIII (1965), 181-189.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2413, "problem_number": "EP-963", "title": "Erdős Problem #963", "statement": "Let $f(n)$ be the maximal $k$ such that in any set $A\\subset \\mathbb{R}$ of size $n$ there is a subset $B\\subseteq A$ of size $\\lvert B\\rvert\\geq k$ which is dissociated that is, the sums $\\sum_{b\\in S}b$ are distinct for all $S\\subseteq B$. Estimate $f(n)$ - in particular, is it true that $ f(n)\\geq \\lfloor \\log_2 n\\rfloor? $ ", "background": "Erdos noted that the greedy algorithm showed $f(n)\\geq \\lfloor \\log_3 n\\rfloor$.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2414, "problem_number": "EP-968", "title": "Erdős Problem #968", "statement": "Let $u_n=p_n/n$, where $p_n$ is the $n$th prime. Does the set of $n$ such that $u_nu_{n+1}$ has positive density.\nErdos also asks whether $ u_nu_{n+1}>u_{n+2} $ have infinitely many solutions.\nReferences\n\n\n[ErPr61] Erdos, P. and Prachar, K., S\"{a}tze und {P}robleme \"{u}ber {$p\\sb{k}/k$}. Abh. Math. Sem. Univ. Hamburg (1961/62), 251--256.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2415, "problem_number": "EP-969", "title": "Erdős Problem #969", "statement": "Let $Q(x)$ count the number of squarefree integers in $[1,x]$. Determine the order of magnitude in the error term in the asymptotic $ Q(x)=\\frac{6}{\\pi^2}x+E(x). $ ", "background": "It is elementary to prove $E(x)\\ll x^{1/2}$, and the prime number theorem implies $o(x^{1/2})$. The best known unconditional upper bound is of the shape $x^{1/2-o(1)}$, due to Walfisz \\cite{Wa63}. Evelyn and Linfoot \\cite{EvLi31} proved that $ E(x) \\gg x^{1/4}, $ and this is likely the true order of magnitude. The Riemann Hypothesis would follow from $E(x)\\ll x^{1/4}$.\nThe true order of magnitude is unknown even assuming the Riemann Hypothesis. Conditional on this assumption, the best known upper bound is $ E(x)\\ll x^{\\frac{11}{35}+o(1)}, $ due to Liu \\cite{Li16}.\nReferences\n\n\n[EvLi31] Evelyn, C. J. A. and Linfoot, E. H., On a problem in the additive theory of numbers. Ann. of Math. (2) (1931), 261--270.\n\n[Li16] Liu, H.-Q., On the distribution of squarefree numbers. J. Number Theory (2016), 202--222.\n\n[Wa63] Walfisz, Arnold, Weylsche {E}xponentialsummen in der neueren {Z}ahlentheorie. (1963), 231.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2416, "problem_number": "EP-970", "title": "Erdős Problem #970", "statement": "Let $h(k)$ be Jacobsthal's function, defined to as the minimal $m$ such that, if $n$ has at most $k$ prime factors, then in any set of $m$ consecutive integers there exists an integer coprime to $n$. Determine the order of magnitude of $h(k)$. In particular, is it true that $ h(k) \\ll k^2? $ ", "background": "That $h(k)\\ll k^2$ is a conjecture of Jacobsthal. Iwaniec \\cite{Iw78} proved $ h(k) \\ll (k\\log k)^2. $ The best lower bound known is $ h(k) \\gg \\frac{(\\log k)(\\log\\log\\log k)}{(\\log\\log k)^2}k, $ due to Ford, Green, Konyagin, Maynard, and Tao \\cite{FGKMT18}.\nThis is a more general form of the function considered in [687].\nReferences\n\n\n[FGKMT18] Ford, Kevin and Green, Ben and Konyagin, Sergei and Maynard, James and Tao, Terence, Long gaps between primes. J. Amer. Math. Soc. (2018), 65-105.\n\n[Iw78] Iwaniec, Henryk, On the problem of {J}acobsthal. Demonstratio Math. (1978), 225--231.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2417, "problem_number": "EP-971", "title": "Erdős Problem #971", "statement": "Let $p(a,d)$ be the least prime congruent to $a\\pmod{d}$. Does there exist a constant $c>0$ such that, for all large $d$, $ p(a,d) > (1+c)\\phi(d)\\log d $ for $\\gg \\phi(d)$ many values of $a$?", "background": "Erdos \\cite{Er49c} could prove this is true for an infinite sequence of $d$. He also proved that, for any $\\epsilon>0$, $ p(a,d)< \\epsilon \\phi(d)\\log d $ for $\\gg_\\epsilon \\phi(d)$ many values of $a$.\nReferences\n\n\n[Er49c] Erdos, P., On some applications of {B}run's method. Acta Univ. Szeged. Sect. Sci. Math. (1949), 57--63.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2418, "problem_number": "EP-972", "title": "Erdős Problem #972", "statement": "Let $\\alpha>1$ be irrational. Are there infinitely many primes $p$ such that $\\lfloor p\\alpha\\rfloor$ is also prime?", "background": "Vinogradov \\cite{Vi48} proved that the sequence $\\{p\\alpha\\}$ is uniformly distributed for every irrational $\\alpha$, and hence there are infinitely many primes $p$ of the shape $p=\\lfloor n\\alpha\\rfloor$ for every irrational $\\alpha>1$. Indeed, this occurs if and only if $ \\frac{p}{\\alpha}\\leq n<\\frac{p+1}{\\alpha}, $ which is true if and only if $\\{p\\alpha^{-1}\\}>1-\\alpha^{-1}$, which happens infinitely often by the uniform distribution of $\\{p\\alpha^{-1}\\}$.\nReferences\n\n\n[Vi48] Vinogradov, I. M., On an estimate of trigonometric sums with prime numbers. Izv. Akad. Nauk SSSR Ser. Mat. (1948), 225--248.\",\n \"difficulty\": \"L3\"\n},{", "difficulty_level_id": 3, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2419, "problem_number": "EP-973", "title": "Erdős Problem #973", "statement": "Does there exist a constant $C>1$ such that, for every $n\\geq 2$, there exists a sequence $z_i\\in \\mathbb{C}$ with $z_1=1$ and $\\lvert z_i\\rvert \\geq 1$ for all $1\\leq i\\leq n$ with $ \\max_{2\\leq k\\leq n+1}\\left\\lvert \\sum_{1\\leq i\\leq n}z_i^k\\right\\rvert < C^{-n}? $ ", "background": "This is Problem 7.3 in \\cite{Ha74}, where it is attributed to Erdos.\nErdos proved (as described on p.35 of \\cite{Tu84b}) that such a sequence does exist with $\\lvert z_i\\rvert\\leq 1$. Indeed, Erdos' construction gives a value of $C\\approx 1.32$.\nIn \\cite{Er92f} (a different) Erdos refines this analysis, proving that if $ M_2=\\min_{z_j} \\max_{2\\leq k\\leq n+1} \\left\\lvert \\sum_{1\\leq j\\leq n}z_j^k\\right\\rvert, $ where the minimum is take over all $z_j\\in \\mathbb{C}$ with $\\max \\lvert z_j\\rvert=1$, then $ (1.746)^{-n} < M_2 < (1.745)^{-n}. $ Tang notes in the comments that Theorem 6.1 of \\cite{Tu84b} implies that, if $\\lvert z_i\\rvert \\geq 1$ for all $i$, then $ \\max_{2\\leq k\\leq n+1}\\left\\lvert \\sum_{1\\leq i\\leq n}z_i^k\\right\\rvert \\geq (2e)^{-(1+o(1))n}. $ See also [519].\nReferences\n\n\n[Er92f] Erdos, L., On some problems of {P}. {T}ur\\'an concerning power sums of\ncomplex numbers. Acta Math. Hungar. (1992), 11--24.\n\n[Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155--180.\n\n[Tu84b] Tur\\'an, Paul, On a new method of analysis and its applications. (1984), xvi+584.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2420, "problem_number": "EP-975", "title": "Erdős Problem #975", "statement": "Let $f\\in \\mathbb{Z}[x]$ be an irreducible non-constant polynomial such that $f(n)\\geq 1$ for all large $n\\in\\mathbb{N}$. Does there exist a constant $c=c(f)>0$ such that $ \\sum_{n\\leq X} \\tau(f(n))\\sim cX\\log X, $ where $\\tau$ is the divisor function?", "background": "Van der Corput \\cite{Va39} proved that $ \\sum_{n\\leq X} \\tau(f(n))\\gg_f X\\log X. $ Erdos \\cite{Er52b} proved using elementary methods that $ \\sum_{n\\leq X} \\tau(f(n))\\ll_f X\\log X. $ Such an asymptotic formula is known whenever $f$ is an irreducible quadratic, as proved by Hooley \\cite{Ho63}. The form of $c$ depends on $f$ in a complicated fashion (see the work of McKee \\cite{Mc95}, \\cite{Mc97}, and \\cite{Mc99} for expressions for various types of quadratic $f$). For example, $ \\sum_{n\\leq x}\\tau(n^2+1)=\\frac{3}{\\pi}x\\log x+O(x). $ Tao has a blog post on this topic.\nReferences\n\n\n[Er52b] Erdos, P., On the sum {$\\sum^x_{k=1} d(f(k))$}. J. London Math. Soc. (1952), 7--15.\n\n[Ho63] Hooley, Christopher, On the number of divisors of a quadratic polynomial. Acta Math. (1963), 97--114.\n\n[Mc95] McKee, James, On the average number of divisors of quadratic polynomials. Math. Proc. Cambridge Philos. Soc. (1995), 389--392.\n\n[Mc97] McKee, James, A note on the number of divisors of quadratic polynomials. (1997), 275--281.\n\n[Mc99] McKee, James, The average number of divisors of an irreducible quadratic\npolynomial. Math. Proc. Cambridge Philos. Soc. (1999), 17--22.\n\n[Va39] van der Corput, J. G., Une in\\'{e}galit\\'{e}{} relative au nombre des diviseurs. Nederl. Akad. Wetensch., Proc. (1939), 547--553.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2421, "problem_number": "EP-976", "title": "Erdős Problem #976", "statement": "Let $f\\in \\mathbb{Z}[x]$ be an irreducible polynomial of degree $d\\geq 2$. Let $F_f(n)$ be maximal such that there exists $1\\leq m\\leq n$ with $f(m)$ is divisible by a prime $\\geq F_f(n)$. Equivalently, $F_f(n)$ is the greatest prime divisor of $ \\prod_{1\\leq m\\leq n}f(m). $ Estimate $F_f(n)$. In particular, is it true that $F_f(n)\\gg n^{1+c}$ for some constant $c>0$? Or even $\\gg n^d$?", "background": "Nagell and Ricci \\cite{Na22} proved that $ F_f(n) \\gg n\\log n, $ which Erdos \\cite{Er52c} improved to $ F_f(n) \\gg n(\\log n)^{\\log\\log\\log n}. $ In \\cite{Er65b} he claimed a proof of $ F_f(n) \\gg n\\exp((\\log n)^c) $ for some constant $c>0$, but said he had never published the proof, which was 'fairly complicated'. This seems to have been flawed, since Erdos and Schinzel \\cite{ErSc90} later published a weaker bound. A proof of the stronger bound above was finally provided by Tenenbaum \\cite{Te90}.\nReferences\n\n\n[Er52c] Erdos, P., On the greatest prime factor of {$\\prod^x_{k=1}f(k)$}. J. London Math. Soc. (1952), 379--384.\n\n[Er65b] Erdos, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244.\n\n[ErSc90] Erdos, P. and Schinzel, A., On the greatest prime factor of {$\\prod^x_{k=1}f(k)$}. Acta Arith. (1990), 191--200.\n\n[Na22] No reference found.\n\n\n[Te90] Tenenbaum, G\\'{e}rald, Sur une question d'{E}rd\\H{o}s et {S}chinzel. (1990), 405--443.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2422, "problem_number": "EP-978", "title": "Erdős Problem #978", "statement": "Let $f\\in \\mathbb{Z}[x]$ be an irreducible polynomial of degree $k>2$ (and suppose that $k\neq 2^l$ for any $l\\geq 1$) such that the leading coefficient of $f$ is positive.\nDoes the set of integers $n\\geq 1$ for which $f(n)$ is $(k-1)$-power-free have positive density?\nAre there infinitely many $n$ for which $f(n)$ is $(k-2)$-power-free?\nIn particular, does $ n^4+2 $ represent infinitely many squarefree numbers?", "background": "Erdos \\cite{Er53} proved there are infinitely many $n$ for which $f(n)$ is $(k-1)$-power-free, except for possibly when $k=2^l$, when it may happen that $2^{l-1}\\mid f(n)$ for all $n$.\nHooley \\cite{Ho67} settled the first question, in fact providing a precise asymptotic for the number of such $n\\leq x$.\nHeath-Brown \\cite{He06} proved the answer to the second question is yes when $k\\geq 10$, and Browning \\cite{Br11} extended this to $k\\geq 9$ (in fact establishing an asymptotic formula for the number of such $n$).\nIn \\cite{Er65b} Erdos mentions the question of whether $2^n\\pm 1$ represents infinitely many $k$th power-free integers, or $n!\\pm 1$, but that these are 'intractable at present'. (See also [936].)\nReferences\n\n\n[Br11] Browning, T. D., Power-free values of polynomials. Arch. Math. (Basel) (2011), 139--150.\n\n[Er53] Erdos, P., Arithmetical properties of polynomials. J. London Math. Soc. (1953), 416--425.\n\n[Er65b] Erdos, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244.\n\n[He06] Heath-Brown, D. R., Counting rational points on algebraic varieties. (2006), 51--95.\n\n[Ho67] Hooley, C., On the power free values of polynomials. Mathematika (1967), 21--26.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2423, "problem_number": "EP-979", "title": "Erdős Problem #979", "statement": "Let $k\\geq 2$, and let $f_k(n)$ count the number of solutions to $ n=p_1^k+\\cdots+p_k^k, $ where the $p_i$ are prime numbers. Is it true that $\\limsup f_k(n)=\\infty$?", "background": "Erdos \\cite{Er37b} proved this is true when $k=2$, and also when $k=3$ (but this proof appears to be unpublished).\nReferences\n\n\n[Er37b] Erdos, Paul, On the {S}um and {D}ifference of {S}quares of {P}rimes. J. London Math. Soc. (1937), 133--136.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2424, "problem_number": "EP-983", "title": "Erdős Problem #983", "statement": "Let $n\\geq 2$ and $\\pi(n)r$ many $a\\in A$ are only divisible by primes from $\\{p_1,\\ldots,p_r\\}$.\nIs it true that $ 2\\pi(n^{1/2})-f(\\pi(n)+1,n)\\to \\infty $ as $n\\to \\infty$?\nIn general, estimate $f(k,n)$, particularly when $\\pi(n)+10$ and, for any constant $1>c>0$, $ f(cn,n)=\\log\\log n+(c_1+o(1))\\sqrt{2\\log\\log n}, $ where $ c=\\frac{1}{\\sqrt{2\\pi}}\\int_{-\\infty}^{c_1}e^{-x^2/2}\\mathrm{d}x. $ \nReferences\n\n\n[Er70b] Erdos, P., Some applications of graph theory to number theory. Proc. Second Chapel Hill Conf. on Combinatorial Mathematics and its Applications (Univ. North Carolina, Chapel Hill, N.C., 1970) (1970), 136-145.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2425, "problem_number": "EP-985", "title": "Erdős Problem #985", "statement": "Is it true that, for every prime $p$, there is a prime $q0$.", "background": "Spencer \\cite{Sp77} proved this for $k=3$ and Mattheus and Verstraete \\cite{MaVe23} proved this for $k=4$.\nThe best general bounds available are $ \\frac{n^{\\frac{k+1}{2}}}{(\\log n)^{\\frac{1}{k-2}-\\frac{k+1}{2}}}\\ll_k R(k,n) \\ll_k \\frac{n^{k-1}}{(\\log n)^{k-2}}. $ The lower bound was proved by Bohman and Keevash \\cite{BoKe10}. The upper bound was proved by Ajtai, Koml\\'{o}s, and Szemer\\'{e}di \\cite{AKS80}. Li, Rousseau, and Zang \\cite{LRZ01} have shown that $\\ll_k$ in the upper bound can be improved to $\\leq (1+o(1))$.\nThe special case $k=3$ is the topic of [165] and $k=4$ is the topic of [166].\nThis problem is #6 in Ramsey Theory in the graphs problem collection.\nSee also [920].\nReferences\n\n\n[AKS80] Ajtai, Mikl\\'{o}s and Koml\\'{o}s, J\\'{a}nos and Szemer\\'{e}di, Endre, A note on Ramsey numbers. J. Combin. Theory Ser. A (1980), 354-360.\n\n[BoKe10] Bohman, Tom and Keevash, Peter, The early evolution of the {$H$}-free process. Invent. Math. (2010), 291--336.\n\n[LRZ01] Li, Yusheng and Rousseau, Cecil C. and Zang, Wenan, Asymptotic upper bounds for {R}amsey functions. Graphs Combin. (2001), 123--128.\n\n[MaVe23] Mattheus, S. and Verstraete, J., The asymptotics of $r(4,t)$. arXiv:2306.04007 (2023).\n\n[Sp77] Spencer, J., Asymptotic lower bounds for Ramsey functions. Discrete Math. (1977), 69-76.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2427, "problem_number": "EP-987", "title": "Erdős Problem #987", "statement": "Let $x_1,x_2,\\ldots \\in (0,1)$ be an infinite sequence and let $ A_k=\\limsup_{n\\to \\infty}\\left\\lvert \\sum_{j\\leq n} e(kx_j)\\right\\rvert, $ where $e(x)=e^{2\\pi ix}$.\nIs it true that $ \\limsup_{k\\to \\infty} A_k=\\infty? $ Is it possible for $A_k=o(k)$?", "background": "This is Problem 7.21 in \\cite{Ha74}, where it is attributed to Erdos.\nErdos \\cite{Er64b} remarks it is 'easy to see' that $ \\limsup_{k\\to \\infty}\\left(\\sup_n\\left\\lvert \\sum_{j\\leq n} e(kx_j)\\right\\rvert\\right)=\\infty. $ Erdos \\cite{Er65b} later found a 'very easy' proof that $A_k\\gg \\log k$ for infinitely many $k$. Clunie \\cite{Cl67} proved that $A_k\\gg k^{1/2}$ infinitely often, and that there exist sequences with $A_k\\leq k$ for all $k$. Tao has independently found a proof that $A_k\\gg k^{1/2}$ infinitely often (see the comment section).\nLiu \\cite{Li69} showed that, for any $\\epsilon>0$, $A_k\\gg k^{1-\\epsilon}$ infinitely often, under the additional assumption that there are only a finite number of distinct points. Clunie observed in the Mathscinet review of \\cite{Li69}, however, that under this assumption in fact $A_k=\\infty$ infinitely often.\nThe question of whether $A_k=o(k)$ is possible (repeated in \\cite{Er65b} and \\cite{Ha74}) seems to still be open.\nReferences\n\n\n[Cl67] Clunie, J., On a problem of {E}rd\\H{o}s. J. London Math. Soc. (1967), 133--136.\n\n[Er64b] Erdos, P., Problems and results on diophantine approximations. Compositio Math. (1964), 52-65.\n\n[Er65b] Erdos, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244.\n\n[Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155--180.\n\n[Li69] Lindstr\"{o}m, B., An inequality for $B_2$-sequences. J. Combinatorial Theory (1969), 211-212.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2428, "problem_number": "EP-990", "title": "Erdős Problem #990", "statement": "Let $f=a_0+\\cdots+a_dx^d\\in \\mathbb{C}[x]$ be a polynomial. Is it true that, if $f$ has roots $z_1,\\ldots,z_d$ with corresponding arguments $\\theta_1,\\ldots,\\theta_d\\in [0,2\\pi]$, then for all intervals $I\\subseteq [0,2\\pi]$ $ \\left\\lvert (\\# \\theta_i \\in I) - \\frac{\\lvert I\\rvert}{2\\pi}d\\right\\rvert \\ll \\left(n\\log M\\right)^{1/2}, $ where $n$ is the number of non-zero coefficients of $f$ and $ M=\\frac{\\lvert a_0\\rvert+\\cdots +\\lvert a_d\\rvert}{(\\lvert a_0\\rvert\\lvert a_d\\rvert)^{1/2}}. $ ", "background": "Erdos and Tur\\'{a}n \\cite{ErTu50} proved such an upper bound with $n$ replaced by $d$.\nReferences\n\n\n[ErTu50] Erdos, P. and Tur\\'an, P., On the distribution of roots of polynomials. Ann. of Math. (2) (1950), 105--119.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2429, "problem_number": "EP-992", "title": "Erdős Problem #992", "statement": "Let $x_1 \\lambda>1$ for all $i$.\nReferences\n\n\n[Ba81] No reference found.\n\n\n[Ca50] Cassels, J. W. S., Some metrical theorems of {D}iophantine approximation. {III}. Proc. Cambridge Philos. Soc. (1950), 219--225.\n\n[ErKo49] Erdos, P. and Koksma, J. F., On the uniform distribution modulo {$1$} of sequences\n{$(f(n,\\theta))$}. Nederl. Akad. Wetensch., Proc. (1949), 851--854 = Indagationes Math. 11, 299--302.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2430, "problem_number": "EP-995", "title": "Erdős Problem #995", "statement": "Let $n_10$, for almost all $\\alpha$ $ \\limsup_{N\\to \\infty}\\frac{1}{N(\\log\\log N)^{\\frac{1}{2}-\\epsilon}}\\sum_{1\\leq k\\leq N}f(\\{\\alpha n_k\\})=\\infty. $ Erdos also proved that, for every lacunary sequence and $f\\in L^2$, for every $\\epsilon>0$, for almost all $\\alpha$, $ \\sum_{1\\leq k\\leq N}\\sum_{1\\leq k\\leq N}f(\\{\\alpha n_k\\})=o( N(\\log N)^{\\frac{1}{2}+\\epsilon}). $ Erdos \\cite{Er64b} thought that his lower bound was closer to the truth.\nReferences\n\n\n[Er49d] Erdos, P., On the strong law of large numbers. Trans. Amer. Math. Soc. (1949), 51--56.\n\n[Er64b] Erdos, P., Problems and results on diophantine approximations. Compositio Math. (1964), 52-65.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2431, "problem_number": "EP-996", "title": "Erdős Problem #996", "statement": "Let $n_10$ such that, if $ \\| f-f_n\\|_2 \\ll \\frac{1}{(\\log\\log\\log n)^{C}} $ then $ \\lim_{N\\to\\infty}\\frac{1}{N}\\sum_{k\\leq N}f(\\{\\alpha n_k\\})=\\int_0^1 f(x)\\mathrm{d}x $ for almost every $\\alpha$?", "background": "Raikov proved the conclusion always holds (for every $f\\in L^2([0,1])$, with no assumption on $\\| f-f_n\\|_2$) if $n_k=a^k$ for some integer $a\\geq 2$. Erdos \\cite{Er64b} also asked whether this is true for $n_k=\\lfloor a^k\\rfloor$ for some $a>1$.\nKac, Salem, and Zygmund \\cite{KSZ48} proved that the conclusion holds if $ \\| f-f_n\\|_2 \\ll \\frac{1}{(\\log n)^{c}} $ for some constant $c>1$. Erdos \\cite{Er49d} proved that the conclusion holds if $ \\| f-f_n\\|_2 \\ll \\frac{1}{(\\log\\log n)^{c}} $ for some constant $c>1$. Matsuyama \\cite{Ma66} improved this to $c>1/2$.\nIn \\cite{Er64b} Erdos asked whether the conclusion holds for all bounded functions $f$ and lacunary sequences $n_k$.\nReferences\n\n\n[Er49d] Erdos, P., On the strong law of large numbers. Trans. Amer. Math. Soc. (1949), 51--56.\n\n[Er64b] Erdos, P., Problems and results on diophantine approximations. Compositio Math. (1964), 52-65.\n\n[KSZ48] Kac, M. and Salem, R. and Zygmund, A., A gap theorem. Trans. Amer. Math. Soc. (1948), 235--243.\n\n[Ma66] Matsuyama, Noboru, On the strong law of large numbers. Tohoku Math. J. (2) (1966), 259--269.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2432, "problem_number": "EP-997", "title": "Erdős Problem #997", "statement": "Call $x_1,x_2,\\ldots \\in (0,1)$ well-distributed if, for every $\\epsilon>0$, if $k$ is sufficiently large then, for all $n>0$ and intervals $I\\subseteq [0,1]$, $ \\lvert \\# \\{ n0$ is an explicit constant.\nReferences\n\n\n[Ke60] Kesten, Harry, Uniform distribution {${\\rm mod}\\,1$}. Ann. of Math. (2) (1960), 445--471.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2434, "problem_number": "EP-1003", "title": "Erdős Problem #1003", "statement": "Are there infinitely many solutions to $\\phi(n)=\\phi(n+1)$, where $\\phi$ is the Euler totient function?", "background": "Erdos \\cite{Er85e} says that, presumably, for every $k\\geq 1$ the equation $ \\phi(n)=\\phi(n+1)=\\cdots=\\phi(n+k) $ has infinitely many solutions.\nErdos, Pomerance, and S\\'{a}rk\"{o}zy \\cite{EPS87} proved that the number of $n\\leq x$ with $\\phi(n)=\\phi(n+1)$ is at most $ \\frac{x}{\\exp((\\log x)^{1/3})}. $ See [946] for the analogous question with the divisor function.\nReferences\n\n\n[EPS87] Erdos, Paul and Pomerance, Carl and S\\'ark\"ozy, Andr\\'as, On locally repeated values of certain arithmetic functions.\n{III}. Proc. Amer. Math. Soc. (1987), 1--7.\n\n[Er85e] Erdos, P., Some problems and results in number theory. Number theory and combinatorics. Japan 1984 (Tokyo,\nOkayama and Kyoto, 1984) (1985), 65-87.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2435, "problem_number": "EP-1004", "title": "Erdős Problem #1004", "statement": "Let $c>0$. If $x$ is sufficiently large then does there exist $n\\leq x$ such that the values of $\\phi(n+k)$ are all distinct for $1\\leq k\\leq (\\log x)^c$, where $\\phi$ is the Euler totient function?", "background": "Erdos, Pomerenace, and S\\'{a}rk\"{o}zy \\cite{EPS87} proved that if $\\phi(n+k)$ are all distinct for $1\\leq k\\leq K$ then $ K \\leq \\frac{n}{\\exp(c(\\log n)^{1/3})} $ for some constant $c>0$.\nSee [945] for the analogous problem with the divisor function.\nReferences\n\n\n[EPS87] Erdos, Paul and Pomerance, Carl and S\\'ark\"ozy, Andr\\'as, On locally repeated values of certain arithmetic functions.\n{III}. Proc. Amer. Math. Soc. (1987), 1--7.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2436, "problem_number": "EP-1005", "title": "Erdős Problem #1005", "statement": "Let $\\frac{a_1}{b_1},\\frac{a_2}{b_2},\\ldots$ be the Farey fractions of order $n\\geq 4$. Let $f(n)$ be the largest integer such that if $1\\leq k0$ such that $f(n)=(c+o(1))n$ for all large $n$?", "background": "The function $f(n)$ was first considered by Mayer \\cite{Ma42}, who proved $f(n)\\to \\infty$ as $n\\to \\infty$. Erdos \\cite{Er43} proved $f(n)\\gg n$.\nvan Doorn \\cite{vD25b} has proved that $ \\left(\\frac{1}{12}-o(1)\\right)n\\leq f(n) \\leq \\frac{1}{4}n+O(1), $ and conjectures that the upper bound is optimal.\nReferences\n\n\n[Er43] Erdos, P., A note on {F}arey series. Quart. J. Math. Oxford Ser. (1943), 82--85.\n\n[Ma42] Mayer, A. E., A mean value theorem concerning {F}arey series. Quart. J. Math. Oxford Ser. (1942), 48--57.\n\n[vD25b] W. van Doorn, Improved bounds for the Mayer-Erdos phenomenon on similarly ordered Farey fractions. arXiv:2509.00121 (2025).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2437, "problem_number": "EP-1011", "title": "Erdős Problem #1011", "statement": "Let $f_r(n)$ be minimal such that every graph on $n$ vertices with $\\geq f_r(n)$ edges and chromatic number $\\geq r$ contains a triangle. Determine $f_r(n)$.", "background": "Tur\\'{a}n's theorem implies $f_2(n)=\\lfloor n^2/4\\rfloor+1$. Erdos and Gallai \\cite{Er62d} proved $f_3(n)=\\lfloor \\frac{1}{4}(n-1)^2\\rfloor+2$.\nSimonovits showed in his PhD thesis (see the discussion on p. 358 of \\cite{Si74}) that $ f_r(n)=\\frac{n^2}{4}-\\frac{g(r)}{2}{n}+O(1), $ where $g(r)$ is the largest $m$ such that, for any triangle-free graph with chromatic number $\\geq r$, at least $m$ vertices of $G$ need to be removed to obtain a bipartite graph. Simonovits \\cite{Si74} notes $ \\frac{\\log r}{\\log\\log r}r^2 \\ll g(r) \\ll (\\log r)^2r^2. $ Hunter in the comments has noted that other results imply $g(r)\\asymp r^2\\log r$ - in fact $ (1/2-o(1))r^2\\log r\\leq g(r)\\leq (2+o(1))r^2\\log r. $ The lower bound follows from work of Davies and Illingworth \\cite{DaIl22} (see [1104]). The upper bound follows from work of Hefty, Horn, King, and Pfender \\cite{HHKP25} on $R(3,k)$.\nRen, Wang, Wang, and Yang \\cite{RWWY24} showed that, for $n\\geq 150$, $ f_4(n)=\\left\\lfloor\\frac{(n-3)^2}{4}\\right\\rfloor+6. $ \nReferences\n\n\n[DaIl22] Davies, Ewan and Illingworth, Freddie, The {$\\chi$}-{R}amsey problem for triangle-free graphs. SIAM J. Discrete Math. (2022), 1124--1134.\n\n[Er62d] Erdos, P., On a theorem of {R}ademacher-{T}ur\\'an. Illinois J. Math. (1962), 122--127.\n\n[HHKP25] Z. Hefty, P. Horn, D. King, and F. Pfender, Improving $R(3,k)$ in just two bites. arXiv:2510.19718 (2025).\n\n[RWWY24] S. Ren, J. Wang, S. Wang, and W. Yang, Extremal triangle-free graphs with chromatic number at least four. arXiv:2404.07486 (2024).\n\n[Si74] Simonovits, M., Extermal graph problems with symmetrical extremal graphs.\n{A}dditional chromatic conditions. Discrete Math. (1974), 349--376.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2438, "problem_number": "EP-1013", "title": "Erdős Problem #1013", "statement": "Let $h_3(k)$ be the minimal $n$ such that there exists a triangle-free graph on $n$ vertices with chromatic number $k$. Find an asymptotic for $h_3(k)$, and also prove $ \\lim_{k\\to \\infty}\\frac{h_3(k+1)}{h_3(k)}=1. $ ", "background": "The function $h_3(k)$ is dual to the function $f(n)$ considered in [1104], in that $h_3(k)= n$ if and only if $n$ is minimal such that $f(n)=k$.\nGraver and Yackel \\cite{GrYa68} proved $ h_3(k)\\gg \\frac{\\log k}{\\log\\log k}k^2. $ The bounds for $f(n)$ from [1104] imply $ \\left(\\frac{1}{2}-o(1)\\right)k^2\\log k\\leq h_3(k) \\leq (1+o(1))k^2\\log k. $ See also [920] for a generalisation to $K_r$-free graphs.\nReferences\n\n\n[GrYa68] Graver, Jack E. and Yackel, James, Some graph theoretic results associated with Ramsey's theorem. J. Combinatorial Theory (1968), 125--175.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2439, "problem_number": "EP-1014", "title": "Erdős Problem #1014", "statement": "Let $R(k,l)$ be the Ramsey number, so the minimal $n$ such that every graph on at least $n$ vertices contains either a $K_k$ or an independent set on $l$ vertices.\nProve, for fixed $k\\geq 3$, that $ \\lim_{l\\to \\infty}\\frac{R(k,l+1)}{R(k,l)}=1. $ ", "background": "This is open even for $k=3$.\nSee also [544] for other behaviour of $R(3,k)$, and [1030] for the diagonal version of this question.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2440, "problem_number": "EP-1016", "title": "Erdős Problem #1016", "statement": "Let $h(n)$ be minimal such that there is a graph on $n$ vertices with $n+h(n)$ edges which contains a cycle on $k$ vertices, for all $3\\leq k\\leq n$. Estimate $h(n)$. In particular, is it true that $ h(n) \\geq \\log_2n+\\log_*n-O(1), $ where $\\log_*n$ is the iterated logarithmic function?", "background": "Such graphs are called pancyclic. A problem of Bondy \\cite{Bo71}, who claimed a proof (without details) of $ \\log_2(n-1)-1\\leq h(n) \\leq \\log_2n+\\log_*n+O(1). $ Erdos \\cite{Er71} believed the upper bound is closer to the truth, but could not even prove $h(n)-\\log_2n\\to \\infty$.\nA proof of the above lower bound is provided by Griffin \\cite{Gr13}. The first published proof of the upper bound appears to be in Chapter 4.5 of George, Khodkar, and Wallis \\cite{GKW16}.\nReferences\n\n\n[Bo71] Bondy, J. A., Pancyclic graphs. {I}. J. Combinatorial Theory Ser. B (1971), 80--84.\n\n[Er71] Erdos, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc.\nConf., Oxford, 1969) (1971), 97-109.\n\n[GKW16] George, John C. and Khodkar, Abdollah and Wallis, W. D., Pancyclic and bipancyclic graphs. (2016), xii+108.\n\n[Gr13] S. Griffin, Minimal Pancyclicity. arXiv:1312.0274 (2013).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2441, "problem_number": "EP-1017", "title": "Erdős Problem #1017", "statement": "Let $f(n,k)$ be such that every graph on $n$ vertices and $k$ edges can be partitioned into at most $f(n,k)$ edge-disjoint complete graphs. Estimate $f(n,k)$ for $k>n^2/4$.", "background": "The function $f(n,k)$ is sometimes called the clique partition number.\nErdos, Goodman, and P\\'{o}sa \\cite{EGP66} proved that $f(n,k)\\leq n^2/4$ for all $k$ (and in fact the complete graphs can be taken to be edges and triangles), which is best possible in general, as witnessed for example by a complete bipartite graph. In \\cite{Er71} Erd\\H{o} asks vaguely whether this result can be 'sharpened' for $k>n^2/4$.\nLov\\'{a}sz \\cite{Lo68} proved that every graph on $n$ vertices and $k$ edges is the union of $\\binom{n}{2}-k+t$ complete graphs, where $t$ is maximal such that $t^2-t\\leq \\binom{n}{2}-k$, but without the assumption that the complete graphs are edge disjoint. Lov\\'{a}sz's result is sharp in many cases.\nIf $k>n^2/4$ and the graph contains no $K_4$ then this is equivalent to finding the minimum number of edge disjoint triangles. This special case was also asked about by Erdos. A complete answer was provided by Gy\"{o}ri and Keszegh \\cite{GyKe17}, who proved that every $K_4$-free graph with $n$ vertices and $\\lfloor n^2/4\\rfloor+m$ edges contins $m$ pairwise edge disjoint triangles.\nSee also [184] for an analogous problem decomposing into edges and cycles, and [583] for decomposing into paths. The clique partition problem for chordal graphs is the subject of [81].\nReferences\n\n\n[EGP66] Erdos, Paul and Goodman, A. W. and P\\'{o}sa, Lajos, The representation of a graph by set intersections. Canadian J. Math. (1966), 106-112.\n\n[Er71] Erdos, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc.\nConf., Oxford, 1969) (1971), 97-109.\n\n[GyKe17] Gy\\H{o}ri, Ervin and Keszegh, Bal\\'azs, On the number of edge-disjoint triangles in {$K_4$}-free\ngraphs. Combinatorica (2017), 1113--1124.\n\n[Lo68] Lov\\'{a}sz, L., On covering of graphs. Theory of Graphs (Proc. Colloq., Tihany, 1966) (1968), 231-236.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2442, "problem_number": "EP-1021", "title": "Erdős Problem #1021", "statement": "Is it true that, for every $k\\geq 3$, there is a constant $c_k>0$ such that $ \\mathrm{ex}(n,G_k) \\ll n^{3/2-c_k}, $ where $G_k$ is the bipartite graph between $\\{y_1,\\ldots,y_k\\}$ and $\\{z_1,\\ldots,z_{\\binom{k}{2}}\\}$, with each $z_j$ joined to a unique pair of $y_i$?", "background": "A conjecture of Erdos and Simonovits, who proved (in unpublished work) that in such a result one must have $c_k\\to 0$ as $k\\to \\infty$. Erdos \\cite{Er71} could not even prove whether $\\mathrm{ex}(n,G_k)=o(n^{3/2})$.\nWhen $k=3$ the graph $G_3$ is the $6$-cycle $C_6$, for which Erdos \\cite{Er64c} and Bondy and Simonovits \\cite{BoSi74} proved $\\mathrm{ex}(n,C_6)\\ll n^{7/6}$ (see [572]).\nThe graph $G_k$ is the graph $H_k$ of [926] with the vertex $x$ omitted, and can also be described as the $1$-subdivision of $K_k$.\nThis was proved by Conlon and Lee \\cite{CoLe21}, with a value of $c_k=6^{-k}$. This was improved to $c_k=\\frac{1}{4k-6}$ by Janzer \\cite{Ja19}.\nReferences\n\n\n[BoSi74] Bondy, J. A. and Simonovits, M., Cycles of even length in graphs. J. Combinatorial Theory Ser. B (1974), 97-105.\n\n[CoLe21] Conlon, David and Lee, Joonkyung, On the extremal number of subdivisions. Int. Math. Res. Not. IMRN (2021), 9122--9145.\n\n[Er64c] Erdos, P., Extremal problems in graph theory. Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963) (1964), 29-36.\n\n[Er71] Erdos, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc.\nConf., Oxford, 1969) (1971), 97-109.\n\n[Ja19] Janzer, Oliver, Improved bounds for the extremal number of subdivisions. Electron. J. Combin. (2019), Paper No. 3.3, 6.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2443, "problem_number": "EP-1022", "title": "Erdős Problem #1022", "statement": "Is there a constant $c_t$, where $c_t\\to \\infty$ as $t\\to \\infty$, such that if $\\mathcal{F}$ is a finite family of finite sets, all of size at least $t$, and for every set $X$ there are $0$ such that $ \\lim_k \\frac{R(k+1,k)}{R(k,k)}> 1+c. $ ", "background": "A problem of Erdos and S\\'{o}s, who could not even prove whether $R(k+1,k)-R(k,k)>k^c$ for any $c>1$.\nIt is trivial that $R(k+1,k)-R(k,k)\\geq k-2$. Burr, Erdos, Faudree, and Schelp \\cite{BEFS89} proved $ R(k+1,k)-R(k,k)\\geq 2k-5. $ See also [544] for a similar question concerning $R(3,k)$, and [1014] for the general off-diagonal case.\nReferences\n\n\n[BEFS89] Burr, S. A. and Erdos, P. and Faudree, R. J. and Schelp, R.\nH., On the difference between consecutive {R}amsey numbers. Utilitas Math. (1989), 115--118.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2446, "problem_number": "EP-1032", "title": "Erdős Problem #1032", "statement": "We say that a graph is $4$-chromatic critical if it has chromatic number $4$, and removing any edge decreases the chromatic number to $3$.\nIs there, for arbitrarily large $n$, a $4$-chromatic critical graph on $n$ vertices with minimum degree $\\gg n$?", "background": "In \\cite{Er93} Erdos said he asked this 'more than 20 years ago'.\nDirac gave an example of a $6$-chromatic critical graph with minimum degree $>n/2$. This problem is also open for $5$-chromatic critical graphs.\nSimonovits \\cite{Si72} and Toft \\cite{To72} independently constructed $4$-chromatic critical graphs with minimum degree $\\gg n^{1/3}$. Toft conjectured that a $4$-chromatic critical graph on $n$ vertices has at least $(\\frac{5}{3}+o(1))n$ vertices, and has examples to show this would be the best possible.\nSee also [917] and [944].\nReferences\n\n\n[Er93] Erdos, Paul, Some of my favorite solved and unsolved problems in graph\ntheory. Quaestiones Math. (1993), 333-350.\n\n[Si72] Simonovits, M., On colour-critical graphs. Studia Sci. Math. Hungar. (1972), 67--81.\n\n[To72] Toft, B., Two theorems on critical {$4$}-chromatic graphs. Studia Sci. Math. Hungar. (1972), 83--89.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2447, "problem_number": "EP-1033", "title": "Erdős Problem #1033", "statement": "Let $h(n)$ be such that every graph on $n$ vertices with $>n^2/4$ many edges contains a triangle whose vertices have degrees summing to at least $h(n)$. Estimate $h(n)$. In particular, is it true that $ h(n)\\geq (2(\\sqrt{3}-1)-o(1))n? $ ", "background": "Erdos and Laskar \\cite{ErLa85} proved $ 2(\\sqrt{3}-1)n \\geq h(n) \\geq (1+c)n $ for some $c>0$. The lower bound was improved to $\\frac{21}{16}n$ by Fan \\cite{Fa88}.\nReferences\n\n\n[ErLa85] Erdos, Paul and Laskar, Renu, A note on the size of a chordal subgraph. Congr. Numer. (1985), 81--86.\n\n[Fa88] Fan, Genghua, Degree sum for a triangle in a graph. J. Graph Theory (1988), 249--263.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2448, "problem_number": "EP-1035", "title": "Erdős Problem #1035", "statement": "Is there a constant $c>0$ such that every graph on $2^n$ vertices with minimum degree $>(1-c)2^n$ contains the $n$-dimensional hypercube $Q_n$?", "background": "Erdos \\cite{Er93} says 'if the conjecture is false, two related problems could be asked':\n{UL}\n{LI}Determine or estimate the smallest $m>2^n$ such that every graph on $m$ vertices with minimum degree $>(1-c)2^n$ contains a $Q_n$, and {/LI}\n{LI}For which $u_n$ is it true that every graph on $2^n$ vertices with minimum degree $>2^n-u_n$ contains a $Q_n$.{/LI}\n{/UL}\nSee also [576] for the extremal number of edges that guarantee a $Q_n$.\nReferences\n\n\n[Er93] Erdos, Paul, Some of my favorite solved and unsolved problems in graph\ntheory. Quaestiones Math. (1993), 333-350.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2449, "problem_number": "EP-1038", "title": "Erdős Problem #1038", "statement": "Determine the infimum and supremum of $ \\lvert \\{ x\\in \\mathbb{R} : \\lvert f(x)\\rvert < 1\\}\\rvert $ as $f\\in \\mathbb{R}[x]$ ranges over all non-constant monic polynomials, all of whose roots are real and in the interval $[-1,1]$.", "background": "A problem of Erdos, Herzog, and Piranian \\cite{EHP58}, who proved that the measure of the set in question is always at most $2\\sqrt{2}$ under the assumption that all the roots are in $\\{-1,1\\}$, and conjecture this is the best possible upper bound.\nThey also note that the infimum of the set in question is less than $2$, as witnessed by $f(x)=(x+1)(x-1)^m$ for $m\\geq 3$. They further note that if the roots are restricted to $[-2,2]$ then the infimum is zero, as witnessed by a small perturbation of the Chebyshev polynomials.\nThey further conjectured that, if the roots are restricted to $[-2,2]$, then $ \\lvert \\{ x\\in \\mathbb{R} : \\lvert f(x)\\rvert < 1\\}\\rvert\\geq n^{-c} $ for an absolute constant $c>0$. This was proved by Pommerenke \\cite{Po61}, who in fact showed that this set must contain an interval of width $\\gg n^{-4}$.\nThe current best known bounds (see the discussion in the comments) are $ 1.519\\approx 2^{4/3}-1\\leq \\inf \\leq 1.835\\cdots $ and $ \\sup = 2\\sqrt{2}\\approx 2.828. $ \nReferences\n\n\n[EHP58] Erdos, P. and Herzog, F. and Piranian, G., Metric properties of polynomials. J. Analyse Math. (1958), 125-148.\n\n[Po61] Pommerenke, Ch., On metric properties of complex polynomials. Michigan Math. J. (1961), 97-115.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2450, "problem_number": "EP-1039", "title": "Erdős Problem #1039", "statement": "Let $f(z)=\\prod_{i=1}^n(z-z_i)\\in \\mathbb{C}[z]$ with $\\lvert z_i\\rvert \\leq 1$ for all $i$. Let $\\rho(f)$ be the radius of the largest disc which is contained in $\\{z: \\lvert f(z)\\rvert< 1\\}$.\nDetermine the behaviour of $\\rho(f)$. In particular, is it always true that $\\rho(f)\\gg 1/n$?", "background": "A problem of Erdos, Herzog, and Piranian, who note that $f(z)=z^n-1$ has $\\rho(f) \\leq \\frac{\\pi/2}{n}$.\nPommerenke \\cite{Po61} proved that $ \\rho(f) \\geq \\frac{1}{2en^2}. $ Krishnapur, Lundberg, and Ramachandran \\cite{KLR25} proved $ \\rho(f) \\gg \\frac{1}{n\\sqrt{\\log n}}. $ \nReferences\n\n\n[KLR25] M. Krishnapur, E. Lundberg, and K. Ramachandran, On the area of polynomial lemniscates. arXiv:2503.18270 (2025).\n\n[Po61] Pommerenke, Ch., On metric properties of complex polynomials. Michigan Math. J. (1961), 97-115.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2451, "problem_number": "EP-1040", "title": "Erdős Problem #1040", "statement": "Let $F\\subseteq \\mathbb{C}$ be a closed infinite set, and let $\\mu(F)$ be the infimum of $ \\lvert \\{ z: \\lvert f(z)\\rvert < 1\\}\\rvert, $ as $f$ ranges over all polynomials of the shape $\\prod (z-z_i)$ with $z_i\\in F$.\nIs $\\mu(F)$ determined by the transfinite diameter of $F$? In particular, is $\\mu(F)=0$ whenever the transfinite diameter of $F$ is $\\geq 1$?", "background": "A problem of Erdos, Herzog, and Piranian \\cite{EHP58}, who show that the answer is yes if $F$ is a line segment or disc, and that if the transfinite diameter is $<1$ then $\\{ z: \\lvert f(z)\\rvert < 1\\}$ always contains a disc of radius $\\gg_F 1$.\nErdos and Netanyahu \\cite{ErNe73} proved that if $F$ is also bounded and connected, with transfinite diameter $01$ be a rational number. Is $ \\sum_{n=1}^\\infty\\frac{1}{t^n-1}=\\sum_{n=1}^\\infty \\frac{\\tau(n)}{t^n} $ irrational, where $\\tau(n)$ counts the divisors of $n$?", "background": "A conjecture of Chowla. Erdos \\cite{Er48} proved that this is true if $t\\geq 2$ is an integer.\nReferences\n\n\n[Er48] Erdos, P., On arithmetical properties of Lambert series. J. Indian Math. Soc. (N.S.) (1948), 63-66.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2453, "problem_number": "EP-1051", "title": "Erdős Problem #1051", "statement": "Is it true that if $a_11 $ then $ \\sum_{n=1}^\\infty \\frac{1}{a_na_{n+1}} $ is irrational?", "background": "In \\cite{Er88c} Erdos notes this is true if $a_n\\to \\infty$ 'rapidly'.\nReferences\n\n\n[Er88c] Erd\"{o}s, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2454, "problem_number": "EP-1052", "title": "Erdős Problem #1052", "statement": "A unitary divisor of $n$ is $d\\mid n$ such that $(d,n/d)=1$. A number $n\\geq 1$ is a unitary perfect number if it is the sum of its unitary divisors (aside from $n$ itself).\nAre there only finite many unitary perfect numbers?", "background": "Guy \\cite{Gu04} reports that Carlitz, Erdos, and Subbarao offer \\$10 for settling this question, and that Subbarao offers 10 cents for each new example.\nThere are no odd unitary perfect numbers. There are five known unitary perfect numbers (A002827 in the OEIS): $ 6, 60, 90, 87360, 146361946186458562560000. $ This is problem B3 in Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2455, "problem_number": "EP-1053", "title": "Erdős Problem #1053", "statement": "Call a number $k$-perfect if $\\sigma(n)=kn$, where $\\sigma(n)$ is the sum of the divisors of $n$. Must $k=o(\\log\\log n)$?", "background": "A question of Erdos, as reported in problem B2 of Guy's collection \\cite{Gu04}. Guy further writes 'It has even been suggested that there may be only finitely many $k$-perfect numbers with $k\\geq 3$.' The largest $k$ for which a $k$-perfect number has been found is $k=11$ - see this page for more information.\nThese are known as multiply perfect numbers. When $k=2$ this is the definition of a perfect number.\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2456, "problem_number": "EP-1054", "title": "Erdős Problem #1054", "statement": "Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\\geq 1$.\nIs it true that $f(n)=o(n)$? Or is this true only for almost all $n$, and $\\limsup f(n)/n=\\infty$?", "background": "A question of Erdos reported in problem B2 of Guy's collection \\cite{Gu04}. The function $f(n)$ is undefined for $n=2$ and $n=5$, but is likely well-defined for all $n\\geq 6$ (which would follow from a strong form of Goldbach's conjecture).\nThe sequence of values of $f(n)$ is given by A167485 in the OEIS.\nSee also [468].\nThe strong claim that $f(n)=o(n)$ was disproved by Tao in the comments to [468], in which he proves that the upper density of $\\{ n : f(n)\\leq \\delta n\\}$ is $\\ll \\delta^2$.\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2457, "problem_number": "EP-1055", "title": "Erdős Problem #1055", "statement": "A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\\leq r-1$, with equality for at least one prime factor.\nAre there infinitely many primes in each class? If $p_r$ is the least prime in class $r$, then how does $p_r^{1/r}$ behave?", "background": "A classification due to Erdos and Selfridge. It is easy to prove that the number of primes $\\leq n$ in class $r$ is at most $n^{o(1)}$.\nThe sequence $p_r$ begins $2,13,37,73,1021$ (A005113 in the OEIS). Erdos thought $p_r^{1/r}\\to \\infty$, while Selfridge thought it quite likely to be bounded.\nA similar question can be asked replacing $p+1$ with $p-1$.\nThis is problem A18 in Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2458, "problem_number": "EP-1056", "title": "Erdős Problem #1056", "statement": "Let $k\\geq 2$. Does there exist a prime $p$ and consecutive intervals $I_1,\\ldots,I_k$ such that $ \\prod_{n\\in I_i}n \\equiv 1\\pmod{p} $ for all $1\\leq i\\leq k$?", "background": "This is problem A15 in Guy's collection \\cite{Gu04}, where he reports that in a letter in 1979 Erdos observed that $ 3\\cdot 4\\equiv 5\\cdot 6\\cdot 7\\equiv 1\\pmod{11}, $ establishing the case $k=2$. Makowski \\cite{Ma83} found, for $k=3$, $ 2\\cdot 3\\cdot 4\\cdot 5\\equiv 6\\cdot 7\\cdot 8\\cdot 9\\cdot 10\\cdot 11\\equiv 12\\cdot 13\\cdot 14\\cdot 15\\equiv 1\\pmod{17}. $ Noll and Simmons asked, more generally, whether there are solutions to $q_1!\\equiv\\cdots \\equiv q_k!\\pmod{p}$ for arbitrarily large $k$ (with $q_1<\\cdots0$. Pomerance \\cite{Po89} gave a heuristic suggesting that this is the true order of growth, and in fact $ C(x)= x \\exp\\left(-(1+o(1))\\frac{\\log x\\log\\log\\log x}{\\log\\log x}\\right). $ Alford, Granville, and Pomerance \\cite{AGP94} proved that $C(x)\\to \\infty$, and in fact $C(x)>x^{2/7}$ for large $x$. The lower bound $ C(x)> x^{0.33336704} $ was proved by Harman \\cite{Ha08}. This exponent was improved to $0.3389$ by Lichtman \\cite{Li22}.\nKorselt observed that $n$ being a Carmichael number is equivalent to $n$ being squarefree and $p-1\\mid n-1$ for all primes $p\\mid n$.\nThis is discussed in problem A13 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[AGP94] Alford, W. R. and Granville, Andrew and Pomerance, Carl, There are infinitely many {C}armichael numbers. Ann. of Math. (2) (1994), 703--722.\n\n[Er56c] Erdos, P., On pseudoprimes and {C}armichael numbers. Publ. Math. Debrecen (1956), 201--206.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Ha08] Harman, Glyn, Watt's mean value theorem and {C}armichael numbers. Int. J. Number Theory (2008), 241--248.\n\n[Li22] J. D. Lichtman, Primes in arithmetic progressions to large moduli and shifted primes without large prime factors. arXiv:2211.09641 (2022).\n\n[Po89] Pomerance, Carl, Two methods in elementary analytic number theory. (1989), 135--161.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2460, "problem_number": "EP-1059", "title": "Erdős Problem #1059", "statement": "Are there infinitely many primes $p$ such that $p-k!$ is composite for each $k$ such that $1\\leq k!l$, and all the numbers $n-k!$ are composite for $1\\leq k\\leq l$.\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2461, "problem_number": "EP-1060", "title": "Erdős Problem #1060", "statement": "Let $f(n)$ count the number of solutions to $k\\sigma(k)=n$, where $\\sigma(k)$ is the sum of divisors of $k$. Is it true that $f(n)\\leq n^{o(\\frac{1}{\\log\\log n})}$? Perhaps even $\\leq (\\log n)^{O(1)}$?", "background": "This is discussed in problem B11 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2462, "problem_number": "EP-1061", "title": "Erdős Problem #1061", "statement": "How many solutions are there to $ \\sigma(a)+\\sigma(b)=\\sigma(a+b) $ with $a+b\\leq x$, where $\\sigma$ is the sum of divisors function? Is it $\\sim cx$ for some constant $c>0$?", "background": "A question of Erdos reported in problem B15 of Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2463, "problem_number": "EP-1062", "title": "Erdős Problem #1062", "statement": "Let $f(n)$ be the size of the largest subset $A\\subseteq \\{1,\\ldots,n\\}$ such that there are no three distinct elements $a,b,c\\in A$ such that $a\\mid b$ and $a\\mid c$. How large can $f(n)$ be? Is $\\lim f(n)/n$ irrational?", "background": "The example $[m+1,3m+2]$ shows that $f(n)\\geq\\lceil \\frac{2}{3}n\\rceil$. Lebensold \\cite{Le76} has shown that, for large $n$, $ 0.6725 n \\leq f(n) \\leq 0.6736 n. $ This is problem B24 in Guy's collection \\cite{Gu04}.\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Le76] Lebensold, Kenneth, A divisibility problem. Studies in Appl. Math. (1976/77), 291--294.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2464, "problem_number": "EP-1063", "title": "Erdős Problem #1063", "statement": "Let $k\\geq 2$ and define $n_k\\geq 2k$ to be the least value of $n$ such that $n-i$ divides $\\binom{n}{k}$ for all but one $0\\leq i1$. Is it true that $g_d(n) \\gg n/d$ in general? The upper bound $g_d(n) \\ll n/d$ is trivial, considering widely spaced unit simplices.\nSee [1070] for the general estimate of independence number of unit distance graphs.\nReferences\n\n\n[Cs98] Csizmadia, G., On the independence number of minimum distance graphs. Discrete Comput. Geom. (1998), 179--187.\n\n[PaTo96] Pach, J\\'anos and T\\'oth, G\\'{e}za, On the independence number of coin graphs. Geombinatorics (1996), 30--33.\n\n[Po85] Pollack, R., Increasing the minimum distance of a set of points. J. Combin. Theory Ser. A (1985), 450.\n\n[Sw02] Swanepoel, Konrad J., Independence numbers of planar contact graphs. Discrete Comput. Geom. (2002), 649--670.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2467, "problem_number": "EP-1068", "title": "Erdős Problem #1068", "statement": "Does every graph with chromatic number $\\aleph_1$ contain a countable subgraph which is infinitely vertex-connected?", "background": "I do not think this was originally a question of Erdos - it appears in \\cite{BoPi24} as a 'version of the Erdos-Hajnal problem' (which is [1067]).\nI could not in fact find this in the paper of Erdos and Hajnal \\cite{ErHa66}, however, and hence the first place it appears may in fact be in \\cite{BoPi24}. In hindsight this should not have been included as a separate problem, but this has been discovered too late, and so we must leave it here.\nWe say a graph is infinitely (vertex) connected if any two vertices are connected by infinitely many pairwise vertex-disjoint paths.\nSoukup \\cite{So15} constructed a graph with uncountable chromatic number in which every uncountable set is finitely vertex-connected. A simpler construction was given by Bowler and Pitz \\cite{BoPi24}.\nSee also [1067].\nReferences\n\n\n[BoPi24] N. Bowler and M. Pitz, A note on uncountably chromatic graphs. arXiv:2402.05984 (2024).\n\n[ErHa66] Erdos, P. and Hajnal, A., On chromatic number of graphs and set-systems. Acta Math. Acad. Sci. Hungar. (1966), 61-99.\n\n[So15] Soukup, D\\'aniel T., Trees, ladders and graphs. J. Combin. Theory Ser. B (2015), 96--116.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2468, "problem_number": "EP-1070", "title": "Erdős Problem #1070", "statement": "Let $f(n)$ be maximal such that, given any $n$ points in $\\mathbb{R}^2$, there exist $f(n)$ points such that no two are distance $1$ apart. Estimate $f(n)$. In particular, is it true that $f(n)\\geq n/4$?", "background": "In other words, estimate the minimal independence number of a unit distance graph with $n$ vertices. If $\\omega$ is the independence number and $\\chi$ is the chromatic number then $\\omega \\chi\\geq n$, and hence $f(n)\\geq n/\\chi$, where $\\chi$ is the answer to the Hadwiger-Nelson problem [508].\nThe Moser spindle shows $f(n)\\leq \\frac{2}{7}n\\approx 0.285n$. Larman and Rogers \\cite{LaRo72} noted that if $m_1$ is the supremum of the upper densities of measurable subsets of $\\mathbb{R}^2$ which have no unit distance pairs then $ f(n)\\geq m_1n. $ Croft \\cite{Cr67} gave the best-known lower bound of $m_1\\geq 0.22936$ and hence $ 0.22936n \\leq f(n) \\leq \\frac{2}{7}n\\approx 0.285n. $ Ambrus, Csisz\\'{a}rik, Matolcsi, Varga, and Zs\\'{a}mboki \\cite{ACMVZ23} have proved that $m_1\\leq 0.247$, and hence this approach cannot achieve $f(n)\\geq n/4$. See [232] for more on $m_1$.\nMatolcsi, Ruzsa, Varga, and Zs\\'{a}mboki \\cite{MRVZ23} have improved the upper bound to $ f(n) \\leq \\left(\\frac{1}{4}+o(1)\\right)n. $ They conjecture that $m_1=0.22936\\cdots$ (the lower bound of Croft mentioned above) and $f(n)=(1/4+o(1))n$.\nIf we also insist that no two points are distance $<1$ apart then this is problem becomes [1066].\nReferences\n\n\n[ACMVZ23] G. Ambrus, A. Csisz\\'{a}rik, M. Matolcsi, D. Varga, and P. Zs\\'{a}mboki, The density of planar sets avoiding unit distances. arXiv:2207.14179 (2023).\n\n[Cr67] H. T. Croft, Incidence incidents. Eureka (1967), 22-26.\n\n[LaRo72] Larman, D. G. and Rogers, C. A., The realization of distances within sets in Euclidean space. Mathematika (1972), 1-24.\n\n[MRVZ23] M. Matolcsi, I. Z. Ruzsa, D. Varga, and P. Zs\\'{a}mboki, The fractional chromatic number of the plane is at least $4$. arXiv:2311.10069 (2023).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2469, "problem_number": "EP-1071", "title": "Erdős Problem #1071", "statement": "Is there a finite set of unit line segments (rotated and translated copies of $(0,1)$) in the unit square, no two of which intersect, which are maximal with respect to this property?\nIs there a region $R$ with a maximal set of disjoint unit line segments that is countably infinite?", "background": "A question of Erdos and T\\'{o}th. The answer to the first question is yes (which Erdos gave Danzer \\$10 for). There is no prize mentioned in \\cite{Er87b} for the (still open) second question.\nThere are two examples Erdos gives in \\cite{Er87b}, the {IMAGE=1071-one,first} by Danzer, the {IMAGE=1071-two,second} by an unnamed participant.\nIn \\cite{Er87b} he further asks what happens if the unit line segments are rotated/translated copies of $[0,1]$ that are allowed to intersect only at their endpoints.\nReferences\n\n\n[Er87b] Erdos, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Si\\'{o}fok, 1985) (1987), 167-177.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2470, "problem_number": "EP-1072", "title": "Erdős Problem #1072", "statement": "For any prime $p$, let $f(p)$ be the least integer such that $f(p)!+1\\equiv 0\\pmod{p}$.\nIs it true that there are infinitely many $p$ for which $f(p)=p-1$?\nIs it true that $f(p)/p\\to 0$ for almost all $p$?", "background": "Questions formulated by Erdos, Hardy, and Subbarao \\cite{HaSu02}, who believed that the number of $p\\leq x$ for which $f(p)=p-1$ is $o(x/\\log x)$.\nThese are mentioned in problem A2 of Guy's collection.\nReferences\n\n\n[HaSu02] Hardy, G. E. and Subbarao, M. V., A modified problem of Pillai and some related questions. Amer. Math. Monthly (2002), 554--559.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2471, "problem_number": "EP-1073", "title": "Erdős Problem #1073", "statement": "Let $A(x)$ count the number of composite $ur^{-r}$ such that, for any $\\epsilon>0$, if $n$ is sufficiently large, the following holds.\nAny $r$-uniform hypergraph on $n$ vertices with at least $(1+\\epsilon)(n/r)^r$ many edges contains a subgraph on $m$ vertices with at least $c_rm^r$ edges, where $m=m(n)\\to \\infty$ as $n\\to \\infty$.", "background": "Erdos \\cite{Er64f} proved that this is true with $c_r=r^{-r}$ whenever the graph has at least $\\epsilon n^r$ many edges.\nReferences\n\n\n[Er64f] Erdos, P., On extremal problems of graphs and generalized graphs. Israel J. Math. (1964), 183--190.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2474, "problem_number": "EP-1083", "title": "Erdős Problem #1083", "statement": "Let $d\\geq 3$, and let $f_d(n)$ be the minimal $m$ such that every set of $n$ points in $\\mathbb{R}^d$ determines at least $m$ distinct distances. Estimate $f_d(n)$ - in particular, is it true that $ f_d(n)=n^{\\frac{2}{d}-o(1)}? $ ", "background": "A generalisation of the distinct distance problem [89] to higher dimensions. Erdos \\cite{Er46b} proved $ n^{1/d}\\ll_d f_d(n)\\ll_d n^{2/d}, $ the upper bound construction being given by a set of lattice points.\n{UL}\n{LI} Clarkson, Edelsbrunner, Gubias, Sharir, and Welzl \\cite{CEGSW90} proved $f_3(n)\\gg n^{1/2}$.{/LI}\n{LI}Aronov, Pach, Sharir, and Tardos \\cite{APST04} proved $f_d(n)\\gg n^{\\frac{1}{d-90/77}-o(1)}$ for any $d\\geq 3$ (for example, $f_3(n)\\gg n^{0.546}$).{/LI}\n{LI}Solymosi and Vu \\cite{SoVu08} proved $f_3(n) \\gg n^{3/5}$ and $ f_d(n)\\gg_d n^{\\frac{2}{d}-\\frac{c}{d^2}} $ for all $d\\geq 4$ for some constant $c>0$. (The result in their paper for $d=3$ is slightly weaker than stated here, but uses as a black box the bound for distinct distances in $2$ dimensions; we have recorded the consequence of combining their method with the work of Guth and Katz on [89].){/LI}\n{/UL}\nThe function $f_d(n)$ is essentially the inverse of the function $g_d(n)$ considered in [1089] - with our definitions, $g_d(n)>m$ if and only if $f_d(m)0$, which the triangular lattice shows is the best possible up to the value of $c$. In \\cite{Er75f} he speculated that the triangular lattice is exactly the best possible, and in particular $ f_2(3n^2+3n+1)=9n^2+6n. $ Harborth \\cite{Ha74b} proved that $ f_2(n)=\\lfloor 3n-\\sqrt{12n-3}\\rfloor $ for all $n\\geq 2$.\nIn \\cite{Er75f} he claims the existence of $c_1,c_2>0$ such that $ 6n-c_1n^{2/3}< f_3(n) < 6n-c_2n^{2/3}. $ An upper bound of $ f_3(n) < 6n-0.926n^{2/3} $ for all $n\\geq 2$ was proved by Bezdek and Reid \\cite{BeRe13}.\nIn general, it is known that $ (d-o(1))n \\leq f_d(n) \\leq 2^{O(d)}n, $ the lower bound coming from points arranged in an integer grid and the upper bound from the fact that $2^{O(d)}$ many non-intersecting congruent balls can touch a fixed ball (the kissing number problem).\nA recent survey on contact numbers for sphere packings is by Bezdek and Khan \\cite{BeKa18}.\nSee [223] for the analogous problem with maximal distance $1$.\nReferences\n\n\n[BeKa18] No reference found.\n\n\n[BeRe13] Bezdek, K\\'aroly and Reid, Samuel, Contact graphs of unit sphere packings revisited. J. Geom. (2013), 57--83.\n\n[Er46b] Erdos, P., On sets of distances of {$n$} points. Amer. Math. Monthly (1946), 248--250.\n\n[Er75f] Erdos, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108.\n\n[Ha74b] Harborth, Heiko, L\"{o}sung zu Problem 664A. Elem. Math. (1974), 14-15.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2476, "problem_number": "EP-1085", "title": "Erdős Problem #1085", "statement": "Let $f_d(n)$ be minimal such that, in any set of $n$ points in $\\mathbb{R}^d$, there exist at most $f_d(n)$ pairs of points which distance $1$ apart. Estimate $f_d(n)$.", "background": "The most difficult cases are $d=2$ and $d=3$. When $d=2$ this is the unit distance problem [90], and the best known bounds are $ n^{1+\\frac{c}{\\log\\log n}}< f_2(n) \\ll n^{4/3} $ for some constant $c>0$, the lower bound by Erdos \\cite{Er46b} and the upper bound by Spencer, Szemer\\'{e}di, and Trotter \\cite{SST84}.\nWhen $d=3$ the best known bounds are $ n^{4/3}\\log\\log n \\ll f_3(n) \\ll n^{3/2}\\beta(n) $ where $\\beta(n)$ is a very slowly growing function, the lower bound by Erdos \\cite{Er60b} and the upper bound by Clarkson, Edelsbrunner, Guibas, Sharir, and Welzl \\cite{CEGSW90}.\nA construction of Lenz (taking points on orthogonal circles) shows that, for $d\\geq 4$, $ f_d(n)\\geq \\frac{p-1}{2p}n^2-O(1) $ with $p=\\lfloor d/2\\rfloor$. Erdos \\cite{Er60b} showed that the Erdos-Stone theorem implies $ f_d(n) \\leq \\left(\\frac{p-1}{2p}+o(1)\\right)n^2 $ for $d\\geq 4$.\nErdos \\cite{Er67e} determined $f_d(n)$ up to $O(1)$ for all even $d\\geq 4$. Brass \\cite{Br97} determined $f_4(n)$ exactly. Swanepoel \\cite{Sw09} determined $f_d(n)$ exactly for even $d\\geq 6$. For odd $d\\geq 5$ Erdos and Pach \\cite{ErPa90} proved that there exist constants $c_1(d),c_2(d)>0$ such that $ \\frac{p-1}{2p}n^2 +c_1n^{4/3}\\leq f_d(n) \\leq \\frac{p-1}{2p}n^2 +c_2n^{4/3}. $ \nReferences\n\n\n[Br97] Brass, P., On the maximum number of unit distances among {$n$} points in\ndimension four. (1997), 277--290.\n\n[CEGSW90] Clarkson, Kenneth L. and Edelsbrunner, Herbert and Guibas,\nLeonidas J. and Sharir, Micha and Welzl, Emo, Combinatorial complexity bounds for arrangements of curves and\nspheres. Discrete Comput. Geom. (1990), 99--160.\n\n[Er46b] Erdos, P., On sets of distances of {$n$} points. Amer. Math. Monthly (1946), 248--250.\n\n[Er60b] Erdos, P., On sets of distances of {$n$} points in {E}uclidean space. Magyar Tud. Akad. Mat. Kutat\\'o{} Int. K\"ozl. (1960), 165--169.\n\n[Er67e] Erdos, P., On some applications of graph theory to geometry. Canadian J. Math. (1967), 968--971.\n\n[ErPa90] Erdos, P. and Pach, J., Variations on the theme of repeated distances. Combinatorica (1990), 261--269.\n\n[SST84] Spencer, J. and Szemer\\'{e}di, E. and Trotter, Jr., W., Unit distances in the Euclidean plane. Graph theory and combinatorics (Cambridge, 1983) (1984), 293-303.\n\n[Sw09] Swanepoel, Konrad J., Unit distances and diameters in {E}uclidean spaces. Discrete Comput. Geom. (2009), 1--27.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2477, "problem_number": "EP-1086", "title": "Erdős Problem #1086", "statement": "Let $g(n)$ be minimal such that any set of $n$ points in $\\mathbb{R}^2$ contains the vertices of at most $g(n)$ many triangles with the same area. Estimate $g(n)$.", "background": "Equivalently, how many triangles of area $1$ can a set of $n$ points in $\\mathbb{R}^2$ determine? Erdos and Purdy attribute this question to Oppenheim. Erdos and Purdy \\cite{ErPu71} proved $ n^2\\log\\log n \\ll g(n) \\ll n^{5/2}, $ and believed the lower bound to be closer to the truth. The upper bound has been improved a number of times - by Pach and Sharir \\cite{PaSh92}, Dumitrescu, Sharir, and T\\'{o}th \\cite{DST09}, Apfelbaum and Sharir \\cite{ApSh10}, and Apfaulbaum \\cite{Ap13}. The best known bound is $ g(n) \\ll n^{20/9} $ by Raz and Sharir \\cite{RaSh17}.\nErdos and Purdy also ask a similar question about the higher-dimensional generalisations - more generally, let $g_d^{r}(n)$ be minimal such that any set of $n$ points in $\\mathbb{R}^d$ contains the vertices of at most $g_d^{r}(n)$ many $r$-dimensional simplices with the same volume.\nErdos and Purdy \\cite{ErPu71} proved $g_3^2(n) \\ll n^{8/3}$, and Dumitrescu, Sharir, and T\\'{o}th \\cite{DST09} improved this to $g_3^2(n) \\ll n^{2.4286}$.\nErdos and Purdy \\cite{ErPu71} proved $g_6^2(n)\\gg n^3$. Purdy \\cite{Pu74} proved $ g_4^2(n)\\leq g^2_5(n) \\ll n^{3-c} $ for some constant $c>0$. An observation of Oppenheim (using a construction of Lenz) detailed in \\cite{ErPu71} shows that $ g_{2k+2}^k(n)\\geq \\left(\\frac{1}{(k+1)^{k+1}}+o(1)\\right)n^{k+1} $ and Erdos and Purdy conjecture this is the best possible.\nSee also [90] and [755].\nReferences\n\n\n[Ap13] R. Apfelbaum, Geometric Incidences and Repeated Configurations. Ph.D. Dissertation, School of Computer Science, Tel Aviv University (2013).\n\n[ApSh10] Apfelbaum, Roel and Sharir, Micha, An improved bound on the number of unit area triangles. Discrete Comput. Geom. (2010), 753--761.\n\n[DST09] Dumitrescu, Adrian and Sharir, Micha and T\\'oth, Csaba D., Extremal problems on triangle areas in two and three\ndimensions. J. Combin. Theory Ser. A (2009), 1177--1198.\n\n[ErPu71] Erdos, Paul and Purdy, George, Some extremal problems in geometry. J. Combinatorial Theory Ser. A (1971), 246--252.\n\n[PaSh92] Pach, J\\'anos and Sharir, Micha, Repeated angles in the plane and related problems. J. Combin. Theory Ser. A (1992), 12--22.\n\n[Pu74] Purdy, George, Some extremal problems in geometry. Discrete Math. (1974), 305--315.\n\n[RaSh17] Raz, Orit E. and Sharir, Micha, The number of unit-area triangles in the plane: theme and\nvariation. Combinatorica (2017), 1221--1240.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2478, "problem_number": "EP-1087", "title": "Erdős Problem #1087", "statement": "Let $f(n)$ be minimal such that every set of $n$ points in $\\mathbb{R}^2$ contains at most $f(n)$ many sets of four points which are 'degenerate' in the sense that some pair are the same distance apart. Estimate $f(n)$ - in particular, is it true that $f(n)\\leq n^{3+o(1)}$?", "background": "A question of Erdos and Purdy \\cite{ErPu71}, who proved $ n^3\\log n \\ll f(n) \\ll n^{7/2}. $ \nReferences\n\n\n[ErPu71] Erdos, Paul and Purdy, George, Some extremal problems in geometry. J. Combinatorial Theory Ser. A (1971), 246--252.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2479, "problem_number": "EP-1088", "title": "Erdős Problem #1088", "statement": "Let $f_d(n)$ be the minimal $m$ such that any set of $m$ points in $\\mathbb{R}^d$ contains a set of $n$ points such that any two determined distances are distinct. Estimate $f_d(n)$. In particular, is it true that, for fixed $n\\geq 3$, $ f_d(n)=2^{o(d)}? $ ", "background": "It is easy to prove that $f_d(n) \\leq n^{O_d(1)}$. Erdos \\cite{Er75f} claimed that he and Straus proved $f_d(n)\\leq c_n^d$ for some constant $c_n>0$.\nWhen $d=1$ this is the subject of [530], and $f_1(n)\\asymp n^2$.\nWhen $n=3$ this is the subject of [503]. Erdos could prove $f_2(3)=7$ and Croft \\cite{Cr62} proved $f_3(3)=9$. The results described at [503] demonstrate that $f_d(3)=d^2/2+O(d)$.\nReferences\n\n\n[Cr62] Croft, H. T., $9$-point and $7$-point configurations in $3$-space. Proc. London Math. Soc. (3) (1962), 400-424.\n\n[Er75f] Erdos, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2480, "problem_number": "EP-1089", "title": "Erdős Problem #1089", "statement": "Let $g_d(n)$ be minimal such that every collection of $g_d(n)$ points in $\\mathbb{R}^d$ determines at least $n$ many distinct distances. Estimate $g_d(n)$. In particular, does $ \\lim_{d\\to \\infty}\\frac{g_d(n)}{d^{n-1}} $ exist?", "background": "A question of Kelly. Erdos \\cite{Er75f} writes it is 'easy' to see that $g_d(n)\\gg d^{n-1}$. Erdos and Straus proved (in unpublished work mentioned in \\cite{Er75f}) that $ g_d(n) \\leq c^{d^{1-b_n}} $ for some constants $c>0$ and $b_n>0$.\nIt is trivial that $g_1(3)=4$, and easy to see that $g_2(3)=6$. Croft \\cite{Cr62} proved $g_3(3)=7$. The vertices of a $d$-dimensional cube demonstrate that $ g_d(d+1)>2^d. $ The function $g_d(n)$ is essentially the inverse of the function $f_d(n)$ considered in [1083] - with our definitions, $g_d(n)>m$ if and only if $f_d(m)1$?", "background": "A problem of Erdos, Lacampagne, and Selfridge \\cite{ELS88}, that was also asked in the 1986 problem session of West Coast Number Theory (as reported here).\nIn \\cite{ELS93} they prove that if the deficiency exists and is $\\geq 1$ then $n\\ll 2^k\\sqrt{k}$.\nThe following examples are either from \\cite{ELS88} or here. The following have deficiency $1$ (there are $58$ examples with $n\\leq 10^5$): $ \\binom{7}{3},\\binom{13}{4},\\binom{14}{4},\\binom{23}{5},\\binom{62}{6},\\binom{94}{10},\\binom{95}{10}. $ The examples which follow are the only known examples with deficiency $>1$. The following have deficiency $2$: $ \\binom{44}{8},\\binom{74}{10},\\binom{174}{12},\\binom{239}{14},\\binom{5179}{27},\\binom{8413}{28},\\binom{8414}{28},\\binom{96622}{42}. $ The following have deficiency $3$: $ \\binom{46}{10},\\binom{47}{10},\\binom{241}{16},\\binom{2105}{25},\\binom{1119}{27},\\binom{6459}{33}. $ The following has deficiency $4$: $ \\binom{47}{11}. $ The following has deficiency $9$: $ \\binom{284}{28}. $ See also [384] and [1094].\nBarreto in the comments has given a positive answer to the second question, conditional on two (strong) conjectures.\nReferences\n\n\n[ELS88] Erdos, P. and Lacampagne, C. B. and Selfridge, J. L., Prime factors of binomial coefficients and related problems. Acta Arith. (1988), 507--523.\n\n[ELS93] Erdos, P. and Lacampagne, C. B. and Selfridge, J. L., Estimates of the least prime factor of a binomial coefficient. Math. Comp. (1993), 215--224.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2484, "problem_number": "EP-1094", "title": "Erdős Problem #1094", "statement": "For all $n\\geq 2k$ the least prime factor of $\\binom{n}{k}$ is $\\leq \\max(n/k,k)$, with only finitely many exceptions.", "background": "A stronger form of [384] that appears in a paper of Erdos, Lacampagne, and Selfridge \\cite{ELS88}. Erdos observed that the least prime factor is always $\\leq n/k$ provided $n$ is sufficiently large depending on $k$. Selfridge \\cite{Se77} further conjectured that this always happens if $n\\geq k^2-1$, except $\\binom{62}{6}$.\nThe threshold $g(k)$ below which $\\binom{n}{k}$ is guaranteed to be divisible by a prime $\\leq k$ is the subject of [1095].\nMore precisely, in \\cite{ELS88} they conjecture that if $n\\geq 2k$ then the least prime factor of $\\binom{n}{k}$ is $\\leq \\max(n/k,k)$ with the following $14$ exceptions: $ \\binom{7}{3},\\binom{13}{4},\\binom{23}{5},\\binom{14}{4},\\binom{44}{8},\\binom{46}{10},\\binom{47}{10}, $ $ \\binom{47}{11},\\binom{62}{6},\\binom{74}{10},\\binom{94}{10},\\binom{95}{10},\\binom{241}{16},\\binom{284}{28}. $ They also suggest the stronger conjecture that, with a finite number of exceptions, the least prime factor is $\\leq \\max(n/k,\\sqrt{k})$, or perhaps even $\\leq \\max(n/k,O(\\log k))$. Indeed, in \\cite{ELS93} they provide some further computational evidence, and point out it is consistent with what they know that in fact this holds with $\\leq \\max(n/k,13)$, with only $12$ exceptions.\nDiscussed in problem B31 and B33 of Guy's collection \\cite{Gu04} - there Guy credits Selfridge with the conjecture that if $n> 17.125k$ then $\\binom{n}{k}$ has a prime factor $p\\leq n/k$.\nThis is related to [1093], in that the only counterexamples to this conjecture can occur from $\\binom{n}{k}$ with deficiency $\\geq 1$.\nThere is an interesting discussion about this problem on MathOverflow.\nReferences\n\n\n[ELS88] Erdos, P. and Lacampagne, C. B. and Selfridge, J. L., Prime factors of binomial coefficients and related problems. Acta Arith. (1988), 507--523.\n\n[ELS93] Erdos, P. and Lacampagne, C. B. and Selfridge, J. L., Estimates of the least prime factor of a binomial coefficient. Math. Comp. (1993), 215--224.\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[Se77] J. L. Selfridge, Some problems on the prime factors of consecutive integers. Notices Amer. Math. Soc. (1977), A456-457.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2485, "problem_number": "EP-1095", "title": "Erdős Problem #1095", "statement": "Let $g(k)>k+1$ be the smallest $n$ such that all prime factors of $\\binom{n}{k}$ are $>k$. Estimate $g(k)$.", "background": "A question of Ecklund, Erdos, and Selfridge \\cite{EES74}, who proved $ k^{1+c}0$, and conjectured $g(k)0$, due to Konyagin \\cite{Ko99b}.\nErdos, Lacampagne, and Selfridge \\cite{ELS93} write 'it is clear to every right-thinking person' that $g(k)\\geq\\exp(c\\frac{k}{\\log k})$ for some constant $c>0$.\nSorenson, Sorenson, and Webster \\cite{SSW20} give heuristic evidence that $ \\log g(k) \\asymp \\frac{k}{\\log k}. $ See also [1094].\nReferences\n\n\n[EES74] Ecklund, Jr., E. F. and Erdos, P. and Selfridge, J. L., A new function associated with the prime factors of\n{$(\\sp{n}\\sb{k})$}. Math. Comp. (1974), 647--649.\n\n[ELS93] Erdos, P. and Lacampagne, C. B. and Selfridge, J. L., Estimates of the least prime factor of a binomial coefficient. Math. Comp. (1993), 215--224.\n\n[GrRa96] Granville, Andrew and Ramar\\'{e}, Olivier, Explicit bounds on exponential sums and the scarcity of\nsquarefree binomial coefficients. Mathematika (1996), 73--107.\n\n[Ko99b] Konyagin, S. V., Estimates of the least prime factor of a binomial coefficient. Mathematika (1999), 41--55.\n\n[SSW20] Sorenson, Brianna and Sorenson, Jonathan and Webster,\nJonathan, An algorithm and estimates for the {E}rd\\H{o}s-{S}elfridge\nfunction. (2020), 371--385.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2486, "problem_number": "EP-1096", "title": "Erdős Problem #1096", "statement": "Let $10$ is sufficiently small, $x_{k+1}-x_k \\to 0$?", "background": "A problem of Erdos and Jo\\'{o} posed in the 1991 problem session of Great Western Number Theory.\nThey speculate that the threshold may be $q_0$, where $q_0\\approx 1.3247$ is the real root of $x^3=x+1$, and is the smallest Pisot-Vijayaraghavan number.\nIn \\cite{EJK90} Erd\\H{o}, Jo\\'{o}, and Komornik prove that any Pisot-Vijayaraghavan number cannot have this property, and also prove that, for any $10 $ for all $m\\geq 1$, where $x_k^m$ is the set of those numbers which can be written as a finite sum $\\sum_{n\\geq 0}c_nq^n$ for some $c_n\\in \\{0,\\ldots,m\\}$ (so that the sequence in the question is $x_k^1$). Erdos, Jo\\'{o}, and Schnitzer \\cite{EJS96} improved this to show that, if $10. $ \nReferences\n\n\n[Bu96] Bugeaud, Y., On a property of {P}isot numbers and related questions. Acta Math. Hungar. (1996), 33--39.\n\n[EJK90] Erdos, P\\'al and Jo\\'o, Istv\\'an and Komornik, Vilmos, Characterization of the unique expansions\n{$1=\\sum^\\infty_{i=1}q^{-n_i}$} and related problems. Bull. Soc. Math. France (1990), 377--390.\n\n[EJS96] Erdos, P. and Jo\\'o, I. and Schnitzer, F. J., On {P}isot numbers. Ann. Univ. Sci. Budapest. E\"otv\"os Sect. Math. (1996), 95--99.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2487, "problem_number": "EP-1097", "title": "Erdős Problem #1097", "statement": "Let $A$ be a set of $n$ integers. How many distinct $d$ can occur as the common difference of a three-term arithmetic progression in $A$? Are there always $O(n^{3/2})$ many such $d$?", "background": "A problem Erdos posed in the 1989 problem session of Great Western Number Theory.\nHe states that Erdos and Ruzsa gave an explicit construction which achieved $n^{1+c}$ for some $c>0$, and Erdos and Spencer gave a probabilistic proof which achieved $n^{3/2}$, and speculated this may be the best possible.\nIn the comment section, Chan has noticed that this problem is exactly equivalent to a sums-differences question of Bourgain \\cite{Bo99}, introduced as an arithmetic path towards the Kakeya conjecture: find the smallest $c\\in [1,2]$ such that, for any finite sets of integers $A$ and $B$ and $G\\subseteq A\\times B$ we have $ \\lvert A\\overset{G}{-}B\\rvert \\ll \\max(\\lvert A\\rvert,\\lvert B\\rvert, \\lvert A\\overset{G}{+}B\\rvert)^c $ (where, for example, $A\\overset{G}{+}B$ denotes the set of $a+b$ with $(a,b)\\in G$).\nThis is equivalent in the sense that the greatest exponent $c$ achievable for the main problem here is equal to the smallest constant achievable for the sums-differences question. The current best bounds known are thus $ 1.77898\\cdots \\leq c \\leq 11/6 \\approx 1.833. $ The upper bound is due to Katz and Tao \\cite{KaTa99}. The lower bound is due to Lemm \\cite{Le15} (with a very small improvement found by AlphaEvolve \\cite{GGTW25}).\nReferences\n\n\n[Bo99] Bourgain, J., On the dimension of {K}akeya sets and related maximal\ninequalities. Geom. Funct. Anal. (1999), 256--282.\n\n[GGTW25] B. Georgiev, J. G\\'{o}mez-Serrano, T. Tao, and A. Wagner, Mathematical exploration and discovery at scale. arXiv:2511.02864 (2025).\n\n[KaTa99] Katz, Nets Hawk and Tao, Terence, Bounds on arithmetic projections, and applications to the\n{K}akeya conjecture. Math. Res. Lett. (1999), 625--630.\n\n[Le15] Lemm, Marius, New counterexamples for sums-differences. Proc. Amer. Math. Soc. (2015), 3863--3868.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2488, "problem_number": "EP-1100", "title": "Erdős Problem #1100", "statement": "If $1=d_1<\\cdots0$ and sufficiently large $x$, $ \\max_{n \\exp((\\log\\log x)^{2-\\epsilon}). $ Erdos and Simonovits (see \\cite{Er81h}) proved $ (2^{1/2}+o(1))^k < g(k) < (2-c)^k $ for some constant $c>0$.\nReferences\n\n\n[Er81h] Erdos, P., Some problems and results on additive and multiplicative\nnumber theory. Analytic number theory (Philadelphia, Pa., 1980) (1981), 171-182.\n\n[ErHa78] Erdos, P. and Hall, R. R., On some unconventional problems on the divisors of integers. J. Austral. Math. Soc. Ser. A (1978), 479--485.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2489, "problem_number": "EP-1101", "title": "Erdős Problem #1101", "statement": "If $u=\\{u_10$, if $x$ is sufficiently large then $ \\max_{a_k (1+o(1))t_x \\prod_{i}\\left(1-\\frac{1}{u_i}\\right)^{-1}. $ The strong form of [208] is asking whether if $u_i=p_i^2$, the sequence of prime squares, is good.\nReferences\n\n\n[Er81h] Erdos, P., Some problems and results on additive and multiplicative\nnumber theory. Analytic number theory (Philadelphia, Pa., 1980) (1981), 171-182.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2490, "problem_number": "EP-1103", "title": "Erdős Problem #1103", "statement": "Let $A$ be an infinite sequence of integers such that every $n\\in A+A$ is squarefree. How fast must $A$ grow?", "background": "Erdos notes there exists such a sequence which grows exponentially, but does not expect such a sequence of polynomial growth.\nIn \\cite{Er81h} he asked whether there is an infinite sequence of integers $A$ such that, for every $a\\in A$ and prime $p$, if $ a\\equiv t\\pmod{p^2} $ then $1\\leq t 0.24j^{4/3}$ for all $j$, and further that there exists such a sequence (furthermore with squarefree terms) such that $ a_j < \\exp(5j/\\log j) $ for all large $j$. A superior lower bound of $a_j \\gg j^{15/11-o(1)}$ had earlier been found by Konyagin \\cite{Ko04} when considering the finite case [1109].\nThey also obtain further results for the generalisation from squarefree to $k$-free integers, and also replacing $A+A$ with $A\\cup (A+A)\\cup(A+A+A)$.\nSee also [1109] for the finite analogue of this problem.\nReferences\n\n\n[Er81h] Erdos, P., Some problems and results on additive and multiplicative\nnumber theory. Analytic number theory (Philadelphia, Pa., 1980) (1981), 171-182.\n\n[Ko04] Konyagin, S. V., Problems of the set of square-free numbers. Izv. Ross. Akad. Nauk Ser. Mat. (2004), 63--90.\n\n[vDTa25] W. van Doorn and T. Tao, Growth rates of sequences governed by the squarefree properties of its translates. arXiv:2512.01087 (2025).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2491, "problem_number": "EP-1104", "title": "Erdős Problem #1104", "statement": "Let $f(n)$ be the maximum possible chromatic number of a triangle-free graph on $n$ vertices. Estimate $f(n)$.", "background": "The best bounds available are $ (1-o(1))(n/\\log n)^{1/2}\\leq f(n) \\leq (2+o(1))(n/\\log n)^{1/2}. $ The upper bound is due to Davies and Illingworth \\cite{DaIl22}, the lower bound follows from a construction of Hefty, Horn, King, and Pfender \\cite{HHKP25}.\nOne can ask a similar question for the maximum possible chromatic number of a triangle-free graph on $m$ edges. Let this be $g(m)$. Davies and Illingworth \\cite{DaIl22} prove $ g(m) \\leq (3^{5/3}+o(1))\\left(\\frac{m}{(\\log m)^2}\\right)^{1/3}. $ Kim \\cite{Ki95} gave a construction which implies $g(m) \\gg (m/(\\log m)^2)^{1/3}$.\nThe function $f(n)$ is the inverse to the function $h_3(k)$ considered in [1013].\nA generalisation of $f(n)$ is considered in [920].\nReferences\n\n\n[DaIl22] Davies, Ewan and Illingworth, Freddie, The {$\\chi$}-{R}amsey problem for triangle-free graphs. SIAM J. Discrete Math. (2022), 1124--1134.\n\n[HHKP25] Z. Hefty, P. Horn, D. King, and F. Pfender, Improving $R(3,k)$ in just two bites. arXiv:2510.19718 (2025).\n\n[Ki95] Kim, J. H., The Ramsey number $R(3,t)$ has order of magnitude $t^2/\\log t$. Random Structures and Algorithms (1995), 173-207.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2492, "problem_number": "EP-1105", "title": "Erdős Problem #1105", "statement": "The anti-Ramsey number $\\mathrm{AR}(n,G)$ is the maximum possible number of colours in which the edges of $K_n$ can be coloured without creating a rainbow copy of $G$ (i.e. one in which all edges have different colours).\nLet $C_k$ be the cycle on $k$ vertices. Is it true that $ \\mathrm{AR}(n,C_k)=\\left(\\frac{k-2}{2}+\\frac{1}{k-1}\\right)n+O(1)? $ Let $P_k$ be the path on $k$ vertices and $\\ell=\\lfloor\\frac{k-1}{2}\\rfloor$. If $n\\geq k\\geq 5$ then is $\\mathrm{AR}(n,P_k)$ equal to $ \\max\\left(\\binom{k-2}{2}+1, \\binom{\\ell-1}{2}+(\\ell-1)(n-\\ell+1)+\\epsilon\\right) $ where $\\epsilon=1$ if $k$ is odd and $\\epsilon=2$ otherwise?", "background": "A conjecture of Erdos, Simonovits, and S\\'{o}s \\cite{ESS75}, who gave a simple proof that $\\mathrm{AR}(n,C_3)=n-1$. In this paper they announced proofs of the claimed formula for $\\mathrm{AR}(n,P_k)$ for $n\\geq \\frac{5}{4}k+C$ for some large constant $C$, and also for all $n\\geq k$ if $k$ is sufficiently large, but these never appeared.\nSimonovits and S\\'{o}s \\cite{SiSo84} published a proof that the claimed formula for $\\mathrm{AR}(n,P_k)$ is true for $n\\geq ck^2$ for some constant $c>0$.\nA proof of the formula for $\\mathrm{AR}(n,P_k)$ for all $n\\geq k\\geq 5$ has been announced by Yuan \\cite{Yu21}\nReferences\n\n\n[ESS75] Erdos, P. and Simonovits, M. and S\\'os, V. T., Anti-{R}amsey theorems. (1975), 633--643.\n\n[SiSo84] Simonovits, Mikl\\'os and S\\'os, Vera T., On restricted colourings of {$K_n$}. Combinatorica (1984), 101--110.\n\n[Yu21] L.-T. Yuan, The anti-Ramsey number for paths. arXiv:2102.00807 (2021).\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2493, "problem_number": "EP-1106", "title": "Erdős Problem #1106", "statement": "Let $p(n)$ denote the partition function of $n$ and let $F(n)$ count the number of distinct prime factors of $ \\prod_{1\\leq k\\leq n}p(k). $ Does $F(n)\\to \\infty$ with $n$? Is $F(n)>n$ for all sufficiently large $n$?", "background": "Asked by Erdos at Oberwolfach in 1986. Schinzel noted in the Oberwolfach problem book that $F(n)\\to \\infty$ follows from the asymptotic formula for $p(n)$ and a result of Tijdeman \\cite{Ti73}. This is not obvious; details are given in a paper of Erdos and Ivi\\'{c} (see page 69 of \\cite{ErIv90}).\nSchinzel and Wirsing \\cite{ScWi87} have proved $F(n) \\gg \\log n$.\nOno \\cite{On00} has proved that every prime divides $p(n)$ for some $n\\geq 1$ (indeed this holds, for any fixed prime, for a positive density set of $n$).\nReferences\n\n\n[ErIv90] Erdos, Paul and Ivi\\'c, Aleksandar, The distribution of values of a certain class of arithmetic\nfunctions at consecutive integers. (1990), 45--91.\n\n[On00] Ono, Ken, Distribution of the partition function modulo {$m$}. Ann. of Math. (2) (2000), 293--307.\n\n[ScWi87] Schinzel, A. and Wirsing, E., Multiplicative properties of the partition function. Proc. Indian Acad. Sci. Math. Sci. (1987), 297--303.\n\n[Ti73] Tijdeman, R., On integers with many small prime factors. Compositio Math. (1973), 319--330.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2494, "problem_number": "EP-1107", "title": "Erdős Problem #1107", "statement": "Let $r\\geq 2$. A number $n$ is $r$-powerful if for every prime $p$ which divides $n$ we have $p^r\\mid n$. Is every large integer the sum of at most $r+1$ many $r$-powerful numbers?", "background": "Given in the 1986 Oberwolfach problem book as a problem of Erdos and Ivi\\'{c}.\nThis is true when $r=2$, as proved by Heath-Brown \\cite{He88} (see [941]).\nSee [940] for the problem of which integers are the sum of at most $r$ many $r$-powerful numbers.\nReferences\n\n\n[He88] Heath-Brown, D. R., Ternary quadratic forms and sums of three square-full numbers. (1988), 137--163.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2495, "problem_number": "EP-1108", "title": "Erdős Problem #1108", "statement": "Let $ A = \\left\\{ \\sum_{n\\in S}n! : S\\subset \\mathbb{N}\\textrm{ finite}\\right\\}. $ If $k\\geq 2$, then does $A$ contain only finitely many $k$th powers? Does it contain only finitely many powerful numbers?", "background": "Asked by Erdos at Oberwolfach in 1988. It is open even whether there are infinitely many squares of the form $1+n!$ (see [398]).\nThis was motivated in part by a problem of Mahler which he discussed with Erdos a few days before his death in 1988: if $k\\geq 5$ and $ A_k= \\left\\{ \\sum_{n\\in S}k^n : S\\subset \\mathbb{N}\\textrm{ finite}\\right\\} $ then does $A_k$ contain only finitely many squares? Mahler showed that there are infinitely many squares in $A_k$ for $k\\leq 4$, and found only one square for $k\\geq 5$, namely $ 1+7+7^2+7^3=400. $ Brindza and Erdos \\cite{BrEr91} proved that, for any $r$, if $n_1!+\\cdots+n_r!$ is powerful then $n_1\\ll_r 1$.\nReferences\n\n\n[BrEr91] Brindza, B. and Erdos, P., On some {D}iophantine problems involving powers and\nfactorials. J. Austral. Math. Soc. Ser. A (1991), 1--7.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2496, "problem_number": "EP-1109", "title": "Erdős Problem #1109", "statement": "Let $f(N)$ be the size of the largest subset $A\\subseteq \\{1,\\ldots,N\\}$ such that every $n\\in A+A$ is squarefree. Estimate $f(N)$. In particular, is it true that $f(N)\\leq N^{o(1)}$, or even $f(N) \\leq (\\log N)^{O(1)}$?", "background": "First studied by Erdos and S\\'{a}rk\"{o}zy \\cite{ErSa87}, who proved $ \\log N \\ll f(N) \\ll N^{3/4}\\log N, $ and guessed the lower bound is nearer the truth. S\\'{a}rk\"{o}zy \\cite{Sa92c} extended this to consider the case of $A+B$ and also looking for sumsets which are $k$-power-free.\nGyarmati \\cite{Gy01} gave an alternative proof of $f(N)\\gg \\log N$, and also gave new bounds for the case of $A+B$. Konyagin \\cite{Ko04} improved this to $ \\log\\log N(\\log N)^2\\ll f(N) \\ll N^{11/15+o(1)}. $ The infinite analogue of this problem is [1103]. (In particular upper bounds for this $f(N)$ directly imply lower bounds for the size of the $a_j$ considered there.)\nReferences\n\n\n[ErSa87] Erdos, P. and S\\'ark\"ozy, A., On divisibility properties of integers of the form {$a+a'$}. Acta Math. Hungar. (1987), 117--122.\n\n[Gy01] Gyarmati, Katalin, On divisibility properties of integers of the form {$ab+1$}. Period. Math. Hungar. (2001), 71--79.\n\n[Ko04] Konyagin, S. V., Problems of the set of square-free numbers. Izv. Ross. Akad. Nauk Ser. Mat. (2004), 63--90.\n\n[Sa92c] S\\'ark\"ozy, G. N., On a problem of {P}. {E}rd\\H{o}s. Acta Math. Hungar. (1992), 271--282.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2497, "problem_number": "EP-1110", "title": "Erdős Problem #1110", "statement": "Let $p>q\\geq 2$ be two coprime integers. We call $n$ representable if it is the sum of integers of the form $p^kq^l$, none of which divide each other.\nIf $\\{p,q\\}\neq \\{2,3\\}$ then what can be said about the density of non-representable numbers? Are there infinitely many coprime non-representable numbers?", "background": "A problem of Erdos and Lewin \\cite{ErLe96}, who proved that there are finitely many non-representable numbers if and only if $\\{p,q\\}=\\{2,3\\}$.\nIndeed, in \\cite{Er92b} Erdos wrote 'last year I made the following silly conjecture': every integer $n$ can be written as the sum of distinct integers of the form $2^k3^l$, none of which divide any other. He wrote 'I mistakenly thought that this was a nice and difficult conjecture but Jansen and several others found a simple proof by induction.'\nThis simple proof is as follows: one proves the stronger fact that such a representation always exists, and moreover if $n$ is even then all the summands can be taken to be even: if $n=2m$ we are done applying the inductive hypothesis to $m$. Otherwise if $n$ is odd then let $3^k$ be the largest power of $3$ which is $\\leq n$ and apply the inductive hypothesis to $n-3^k$ (which is even).\nYu and Chen \\cite{YuCh22} prove that the set of non-representable numbers has density zero whenever $q>3$ or $q=3$ and $p>6$ or $q=2$ and $p>10$. They also prove that there are infinitely many coprime non-representable numbers if $q>3$ or $q=3$ and $p\neq 5$ or $q=2$ and $p\not\\in \\{3,5,9\\}$.\nErdos and Lewin \\cite{ErLe96} also asked whether all large integers $n$ can be written as a sum of $2^k3^l$, none of which divide another, each of which is $>f(n)$ for some $f(n)\\to \\infty$. Let $f(n)$ be the fastest growing such $f(n)$. Yu and Chen \\cite{YuCh22} proved $ \\frac{n}{(\\log n)^{\\log_23}}\\ll f(n) \\ll \\frac{n}{\\log n}. $ Yang and Zhao \\cite{YaZh25} improved the lower bound to $f(n)\\gg n/\\log n$.\nThe case of three powers is the subject of [123], and see also [845] for more on the case $\\{p,q\\}=\\{2,3\\}$. The problem [246] addresses the topic without the non-divisibility condition.\nReferences\n\n\n[Er92b] Erdos, Paul, Some of my favourite problems in various branches of combinatorics. Matematiche (Catania) (1992), 231-240.\n\n[ErLe96] Erdos, P. and Lewin, Mordechai, $d$-complete sequences of integers. Math. Comp. (1996), 837-840.\n\n[YaZh25] Yang, Quan-Hui and Zhao, Lilu, A conjecture of {Y}u and {C}hen related to the {E}rd\\H\nos-{L}ewin theorem. Acta Arith. (2025), 277--286.\n\n[YuCh22] Yu, Wang-Xing and Chen, Yong-Gao, On a conjecture of {E}rd\\H{o}s and {L}ewin. J. Number Theory (2022), 763--778.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2498, "problem_number": "EP-1111", "title": "Erdős Problem #1111", "statement": "If $G$ is a finite graph and $A,B$ are disjoint sets of vertices then we call $A,B$ anticomplete if there are no edges between $A$ and $B$.\nIf $t,c\\geq 1$ then there exists $d\\geq 1$ such that if $\\chi(G)\\geq d$ and $\\omega(G)3$.\nNguyen, Scott, and Seymour \\cite{NSS24} prove that if $t,c\\geq 1$ then there exists $d\\geq 1$ such that if $\\chi(G)\\geq d$ and $\\omega(G)0$, and if $g(p)$ does not have very large values (in a certain technical sense).\nSee also [491].\nReferences\n\n\n[Ma22] Mangerel, Alexander P., Additive functions in short intervals, gaps and a conjecture\nof {E}rd\\H{o}s. Ramanujan J. (2022), 1023--1090.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2504, "problem_number": "EP-1129", "title": "Erdős Problem #1129", "statement": "For $x_1,\\ldots,x_n\\in [-1,1]$ let $ l_k(x)=\\frac{\\prod_{i\neq k}(x-x_i)}{\\prod_{i\neq k}(x_k-x_i)}, $ which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\neq k$.\nDescribe which choice of $x_i$ minimise $ \\Lambda(x_1,\\ldots,x_n)=\\max_{x\\in [-1,1]} \\sum_k \\lvert l_k(x)\\rvert. $ ", "background": "The functions $l_k(x)$ are sometimes called the fundamental functions of Lagrange interpolation, and $\\Lambda$ is sometimes called the Lebesgue constant.\nFaber \\cite{Fa14} proved $ \\Lambda(x_1,\\ldots,x_n)\\gg \\log n $ for all choices of $x_i$, and Bernstein \\cite{Be31} proved it is $>(\\frac{2}{\\pi}-o(1))\\log n$. Erdos \\cite{Er61c} improved this to $ \\Lambda(x_1,\\ldots,x_n)> \\frac{2}{\\pi}\\log n-O(1). $ This is best possible, since taking the $x_i$ as the roots of the $n$th Chebyshev polynomial yields $ \\Lambda(x_1,\\ldots,x_n)< \\frac{2}{\\pi}\\log n+O(1). $ Erdos thought that the minimising choice is characterised by the property that the sums $ \\lambda_i=\\max_{x\\in [x_i,x_{i+1}]}\\sum_k \\lvert l_k(x)\\rvert $ are all equal for $0\\leq i\\leq n$ (where $x_0=-1$ and $x_{n+1}=1$). This conjecture was also made by Bernstein \\cite{Be31}. Kilgore and Cheney \\cite{KiCh76} proved that there exists $x_i$ for which all $\\lambda_i$ are equal. Kilgore \\cite{Ki77} proved that $\\Lambda$ is minimised only when all $\\lambda_i$ are equal. Finally, de Boor and Pinkus \\cite{dBPi78} proved that there exists a unique minimising choice of $x_i$.\nIf $x_1=-1$ and $x_n=1$ then there is a unique minimising set of $x_i$, which are symmetric around $0$. (Such a choice is called canonical.) The exact minimising canonical choice is known only for $n\\leq 4$. For $n=2$ the points are $-1,1$ (with $\\Lambda=1$). For $n=3$ the points are $-1,0,1$ (with $\\Lambda=1.25$), as shown by Bernstein \\cite{Be31}. Rack and Vajda \\cite{RaVa15} have shown that for $n=4$ the points are $-1,-t,t,1$ where $t\\approx 0.4177$ is an explicit algebraic constant (with $\\Lambda \\approx 1.4229$).\nIn \\cite{Er67} Erdos suggests that an easier variant might be to have the $x_i\\in \\mathbb{C}$ with $\\lvert x_i\\rvert=1$, and seek to minimise $\\max_{\\lvert z\\rvert=1}\\sum_{k}\\lvert l_k(z)\\rvert$, adding it 'seems certain' that the minimising $x_i$ are the $n$th roots of unity. This was proved by Brutman \\cite{Br80} for odd $n$ and by Brutman and Pinkus \\cite{BrPi80} for even $n$.\nSee also [671], [1130], and [1132].\nReferences\n\n\n[Be31] S. Bernstein, Sur la limitation des valeurs d'un polynome $P_n(x)$ de degr\\'{e} n sur tout un segment par ses valeurs en $(n+1)$ points du segment. Izv. Akad. Nauk. SSSR (1931), 1025-1050.\n\n[Br80] Brutman, L., On the polynomial and rational projections in the complex\nplane. SIAM J. Numer. Anal. (1980), 366--372.\n\n[BrPi80] Brutman, L. and Pinkus, A., On the {E}rd\\H{o}s conjecture concerning minimal norm\ninterpolation on the unit circle. SIAM J. Numer. Anal. (1980), 373--375.\n\n[Er61c] Erdos, P., Problems and results on the theory of interpolation. II. Acta Math. Acad. Sci. Hungar. (1961), 235-244.\n\n[Er67] Erdos, P., Problems and results on the convergence and divergence properties of the Lagrange interpolation polynomials and some extremal problems. Mathematica (Cluj) (1967), 65-73.\n\n[Fa14] G. Faber, \"{U}ber die interpolatorische Darstellung stetiger Funktionen. Jahresb. der Deutschen Math. Ver. (1914), 190-210.\n\n[Ki77] Kilgore, T. A., Optimization of the norm of the {L}agrange interpolation\noperator. Bull. Amer. Math. Soc. (1977), 1069--1071.\n\n[KiCh76] Kilgore, T. A. and Cheney, E. W., A theorem on interpolation in {H}aar subspaces. Aequationes Math. (1976), 391--400.\n\n[RaVa15] Rack, Heinz-Joachim and Vajda, Robert, Optimal cubic {L}agrange interpolation: extremal node systems\nwith minimal {L}ebesgue constant. Stud. Univ. Babe\\c s-Bolyai Math. (2015), 151--171.\n\n[dBPi78] de Boor, Carl and Pinkus, Allan, Proof of the conjectures of {B}ernstein and {E}rd\\H os\nconcerning the optimal nodes for polynomial interpolation. J. Approx. Theory (1978), 289--303.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 6, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2505, "problem_number": "EP-1130", "title": "Erdős Problem #1130", "statement": "For $x_1,\\ldots,x_n\\in [-1,1]$ let $ l_k(x)=\\frac{\\prod_{i\neq k}(x-x_i)}{\\prod_{i\neq k}(x_k-x_i)}, $ which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\neq k$.\nLet $x_0=-1$ and $x_{n+1}=1$ and $ \\Upsilon(x_1,\\ldots,x_n)=\\min_{0\\leq i\\leq n}\\max_{x\\in[x_i,x_{i+1}]} \\sum_k \\lvert l_k(x)\\rvert. $ Is it true that $ \\Upsilon(x_1,\\ldots,x_n)\\ll \\log n? $ Describe which choice of $x_i$ maximise $\\Upsilon(x_1,\\ldots,x_n)$.", "background": "The functions $l_k(x)$ are sometimes called the fundamental functions of Lagrange interpolation.\nErdos \\cite{Er47} could prove $ \\Upsilon(x_1,\\ldots,x_n)< \\sqrt{n}. $ Erdos thought that the maximising choice is characterised by the property that the sums $ \\lambda_i=\\max_{x\\in [x_i,x_{i+1}]}\\sum_k \\lvert l_k(x)\\rvert $ are all equal for $0\\leq i\\leq n$ (where $x_0=-1$ and $x_{n+1}=1$), which would be the same characterisation as [1129].\nThis is true, and was proved by de Boor and Pinkus \\cite{dBPi78}. It follows by the bounds discussed in [1129] that $ \\Upsilon(x_1,\\ldots,x_n)\\leq \\frac{2}{\\pi}\\log n+O(1). $ See also [1129].\nReferences\n\n\n[Er47] Erdos, P., Some remarks on polynomials. Bull. Amer. Math. Soc. (1947), 1169--1176.\n\n[dBPi78] de Boor, Carl and Pinkus, Allan, Proof of the conjectures of {B}ernstein and {E}rd\\H os\nconcerning the optimal nodes for polynomial interpolation. J. Approx. Theory (1978), 289--303.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2506, "problem_number": "EP-1131", "title": "Erdős Problem #1131", "statement": "For $x_1,\\ldots,x_n\\in [-1,1]$ let $ l_k(x)=\\frac{\\prod_{i\neq k}(x-x_i)}{\\prod_{i\neq k}(x_k-x_i)}, $ which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\neq k$.\nWhat is the minimal value of $ I(x_1,\\ldots,x_n)=\\int_{-1}^1 \\sum_k \\lvert l_k(x)\\rvert^2\\mathrm{d}x? $ In particular, is it true that $ \\min I =2-(1+o(1))\\frac{1}{n}? $ ", "background": "Erdos first conjectured this minimum was achieved by taking the $x_i$ to be the roots of the integral of the Legendre polynomial, since Fejer \\cite{Fe32} had earlier shown these to be minimisers of $ \\max_{x\\in [-1,1]}\\sum_k \\lvert l_k(x)\\rvert^2. $ This was disproved by Szabados \\cite{Sz66} for every $n>3$.\nErdos, Szabados, Varma, and V\\'{e}rtesi \\cite{ESVV94} proved that $ 2-O\\left(\\frac{(\\log n)^2}{n}\\right)\\leq \\min I\\leq 2-\\frac{2}{2n-1} $ where the upper bound is witnessed by the roots of the integral of the Legendre polynomial as above.\nReferences\n\n\n[ESVV94] Erdos, P. and Szabados, J. and Varma, A. K. and V\\'{e}rtesi,\nP., On an interpolation theoretical extremal problem. Studia Sci. Math. Hungar. (1994), 55--60.\n\n[Fe32] Fej\\'{e}r, Leopold, Bestimmung derjenigen {A}bszissen eines {I}ntervalles, f\"ur\nwelche die {Q}uadratsumme der {G}rundfunktionen der\n{L}agrangeschen {I}nterpolation im {I}ntervalle ein\n{M}\"oglichst kleines {M}aximum {B}esitzt. Ann. Scuola Norm. Super. Pisa Cl. Sci. (2) (1932), 263--276.\n\n[Sz66] Szabados, J., On a problem of {P}. {E}rd\\H{o}s. Acta Math. Acad. Sci. Hungar. (1966), 155--157.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2507, "problem_number": "EP-1132", "title": "Erdős Problem #1132", "statement": "For $x_1,\\ldots,x_n\\in [-1,1]$ let $ l_k(x)=\\frac{\\prod_{i\neq k}(x-x_i)}{\\prod_{i\neq k}(x_k-x_i)}, $ which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\neq k$.\nLet $x_1,x_2,\\ldots\\in [-1,1]$ be an infinite sequence, and let $ L_n(x) = \\sum_{1\\leq k\\leq n}\\lvert l_k(x)\\rvert, $ where each $l_k(x)$ is defined above with respect to $x_1,\\ldots,x_n$.\nMust there exist $x\\in (-1,1)$ such that $ L_n(x) >\\frac{2}{\\pi}\\log n-O(1) $ for infinitely many $n$?\nIs it true that $ \\limsup_{n\\to \\infty}\\frac{L_n(x)}{\\log n}\\geq \\frac{2}{\\pi} $ for almost all $x\\in (-1,1)$?", "background": "A result of Bernstein \\cite{Be31} implies that the set of $x\\in(-1,1)$ for which $ \\limsup_{n\\to \\infty}\\frac{L_n(x)}{\\log n}\\geq \\frac{2}{\\pi} $ is everywhere dense.\nErdos \\cite{Er61c} proved that, for any fixed $x_1,\\ldots,x_n\\in [-1,1]$, $ \\max_{x\\in [-1,1]}\\sum_{1\\leq k\\leq n}\\lvert l_k(x)\\rvert>\\frac{2}{\\pi}\\log n-O(1). $ See also [1129] for more on $L_n(x)$.\nReferences\n\n\n[Be31] S. Bernstein, Sur la limitation des valeurs d'un polynome $P_n(x)$ de degr\\'{e} n sur tout un segment par ses valeurs en $(n+1)$ points du segment. Izv. Akad. Nauk. SSSR (1931), 1025-1050.\n\n[Er61c] Erdos, P., Problems and results on the theory of interpolation. II. Acta Math. Acad. Sci. Hungar. (1961), 235-244.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 2, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2508, "problem_number": "EP-1133", "title": "Erdős Problem #1133", "statement": "Let $C>0$. There exists $\\epsilon>0$ such that if $n$ is sufficiently large the following holds.\nFor any $x_1,\\ldots,x_n\\in [-1,1]$ there exist $y_1,\\ldots,y_n\\in [-1,1]$ such that, if $P$ is a polynomial of degree $m<(1+\\epsilon)n$ with $P(x_i)=y_i$ for at least $(1-\\epsilon)n$ many $1\\leq i\\leq n$, then $ \\max_{x\\in [-1,1]}\\lvert P(x)\\rvert >C. $ ", "background": "Erdos proved that, for any $C>0$, there exists $\\epsilon>0$ such that if $n$ is sufficiently large and $m=\\lfloor (1+\\epsilon)n\\rfloor$ then for any $x_1,\\ldots,x_m\\in [-1,1]$ there is a polynomial $P$ of degree $n$ such that $\\lvert P(x_i)\\rvert\\leq 1$ for $1\\leq i\\leq m$ and $ \\max_{x\\in [-1,1]}\\lvert P(x)\\rvert>C. $ The conjectured statement would also imply this, but Erdos in \\cite{Er67} says he could not even prove it for $m=n$.\nReferences\n\n\n[Er67] Erdos, P., Problems and results on the convergence and divergence properties of the Lagrange interpolation polynomials and some extremal problems. Mathematica (Cluj) (1967), 65-73.\",\n \"difficulty\": \"L1\"\n},{", "difficulty_level_id": 1, "status": "open", "category_id": 3, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null }, { "id": 2509, "problem_number": "EP-1135", "title": "Erdős Problem #1135", "statement": "Define $f:\\mathbb{N}\\to \\mathbb{N}$ by $f(n)=n/2$ if $n$ is even and $f(n)=\\frac{3n+1}{2}$ if $n$ is odd.\nGiven any integer $m\\geq 1$ does there exist $k\\geq 1$ such that $f^{(k)}(m)=1$?", "background": "The infamous Collatz conjecture. For a detailed discussion of the history and theory surrounding this problem we refer to the overview by Lagarias \\cite{La10}.\nThis is not a problem due to Erdos; it was first devised by Collatz before 1952. Erdos referred to this problem on several occasions as 'hopeless'. As Lagarias \\cite{La16} notes, the closest Erdos ever came to working on problems of this nature is the theorem described in the remarks to [1134].\nIt is often claimed that Erdos offered \\$500 for a solution to this problem; this claim originated in a survey article by Lagarias \\cite{La85}.\nLagarias reported, in personal communication, that this came from a conversation he had with Erdos and Graham around 1983, in which Graham asked Erdos to make an estimate of what value Erdos would put the problem on his prize scale, to which Erdos replied \\$500. Therefore, strictly speaking, Erdos never offered \\$500 specifically as a prize, but we include this prize value here for comparing those problems which Erdos rated as 'prize problems'.\nThis is Problem E16 in Guy's collection \\cite{Gu04}, in which Guy quotes Erdos as saying \"Mathematics may not be ready for such problems\".\nReferences\n\n\n[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.\n\n[La10] Lagarias, Jeffrey C., The {$3x+1$} problem: an overview. (2010), 3--29.\n\n[La16] Lagarias, Jeffrey C., Erd\\H os, {K}larner, and the {$3x+1$} problem. Amer. Math. Monthly (2016), 753--776.\n\n[La85] Lagarias, Jeffrey C., The {$3x+1$} problem and its generalizations. Amer. Math. Monthly (1985), 3--23.\",\n \"difficulty\": \"L5\"\n}", "difficulty_level_id": 5, "status": "open", "category_id": 1, "set_id": 8, "view_count": 0, "favorite_count": 0, "created_at": "2024-01-01T00:00:00Z", "updated_at": "2024-01-01T00:00:00Z", "published": true, "category": null, "difficulty": null, "set": null } ]