Costas arrays of order 32 and 33
Find a Costas array of order 32 or 33, the smallest orders for which none is known, or extend the complete enumeration of Costas arrays beyond order 29.
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Open problems
Each problem states how progress is verified and what counts as a contribution. Besides the problems curated here, the catalogue includes open conjectures from Formal Conjectures (with Lean statements), optimization constants and the AlphaEvolve problems. Know one that belongs here? Propose a problem.
All fields Mathematics 936Computer science 28Physics & astronomy 22Chemistry & materials 13Life sciences 6Earth & society 6
Combinatorics 158Number theory 500Graph theory 71Algebra 45Geometry 72Analysis 69Probability 11Logic & formalisation 10
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Erdős minimum overlap problem
Improve the numerical upper or lower bounds for the limiting constant in Erdős' minimum overlap problem.
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Hadamard matrices of open orders
Construct Hadamard matrices for orders 4k where none is known, starting with the smallest open orders.
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Large cap sets in F_3^n
Find large subsets of F_3^n with no three points on a line (no x, y, z distinct with x + y + z = 0).
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Optimal Golomb rulers
Find the shortest Golomb ruler (all pairwise mark differences distinct) with n marks. Optimality is proven up to 28 marks (length 585, distributed.net, 2022); 29 marks is the first open case, and shorter rulers for larger n would beat long-standing constructions.
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Small van der Waerden numbers
Determine W(r,k), the least N such that every r-colouring of {1,…,N} contains a monochromatic k-term arithmetic progression. Only seven non-trivial values are known; the open cases W(2,7), W(3,5), W(4,4) and W(5,3) invite better lower-bound colourings and exact computations.
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Smaller covering designs
Improve upper bounds C(v,k,t) for covering designs listed in the La Jolla Covering Repository.
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Snake-in-the-box — longest induced paths in hypercubes
Find the longest induced path (snake) in the n-dimensional hypercube Q_n. Optimal lengths are known only up to n = 8 (98); for n = 9–13 new records were set in 2026 and further improvements are open.
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The sixth Schur number S(6)
Find the largest N such that {1,…,N} can be split into six sum-free sets. After Heule's 2017 SAT proof that S(5) = 160, the best known bound is S(6) ≥ 536, with a large gap to the upper bound.
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Frankl's union-closed sets conjecture
Every finite union-closed family of sets other than {∅} has an element lying in at least half of its sets. Since Gilmer's 2022 entropy breakthrough the best proven fraction is about 0.38; closing the gap to 1/2 is open.
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The 1/3–2/3 conjecture for balanced pairs in posets
Every finite poset that is not a chain has elements x, y such that x precedes y in between 1/3 and 2/3 of its linear extensions. The best general constant is (5−√5)/10 ≈ 0.276; all posets with up to 14 elements have been verified.
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The chromatic number of the plane (Hadwiger–Nelson problem)
Determine how many colours are needed so that no two points of the plane at distance exactly 1 share a colour. The answer is known to be 5, 6 or 7; a concrete sub-goal is a smaller 5-chromatic unit distance graph than the 509-vertex record.
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The Erdős–Rado sunflower conjecture
Show that every family of more than C_k^n sets of size n contains a k-sunflower, for a constant C_k depending only on k. The best bound, about (Ck log n)^n, follows the 2019 breakthrough of Alweiss, Lovett, Wu and Zhang.
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The Erdős–Szekeres happy ending problem
Is every set of 2^{n−2}+1 points in general position in the plane guaranteed to contain n points in convex position? Known exactly up to n = 6 (17 points); the first open case is whether 33 points force a convex 7-gon.
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The lonely runner conjecture
For k+1 runners with distinct constant speeds on a unit circular track, each runner is at some time at distance at least 1/(k+1) from all others. Computer-assisted proofs now cover up to 13 runners; the general case is open.
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4-slope Kakeya-type sum-difference constant
C_3c = SD(\0,1,2,∞\;-1) is the least exponent such that one has the inequality |A stackrelG- B| ≤ max(|A|, |B|, |A stackrelG+ B|, |A stackrelG+ 2B|)^C_3c whenever A, B are finite subsets of reals and G ⊂ A × B, where A stackrelG± rB := a ± rb: a ∈ A, b ∈ B.
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A Sidon set constant
C_5a is the smallest constant such that Sidon sets in \1,…,N\ have cardinality N^1/2 + (C_5a + o(1))N^1/4.
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Asymptotic counting exponent for partial Hadamard matrices
For integers n ≥ 2 and t ≥ 1, an n × t partial Hadamard matrix is a matrix with entries in \± 1\ whose rows are pairwise orthogonal. Let N_n,t denote the number of such matrices. For every fixed n one has N_n,4t = [1+o(1)] A_n,4t qquadas t → ∞, where A_n,4t := 2^4nt+(n-1)^2(8π t)^-n(n-1)/4.
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Davenport constant for C_n^3
In zero-sum theory, the Davenport constant D(G) of a finite abelian group G is defined as the smallest integer l∈ℕ such that every sequence S over G of length lvert Srvert≥ l has a non-empty zero-sum subsequence.
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Difference Bases
For any natural number n, let Δ(n) be the size of the smallest set B of integers such that every natural number from 1 to n is expressible as a difference of two elements of B (such sets are known as difference bases for the interval 1,…,n). Write C(n) := Δ^2(n)/n, and C := inf_n ≥ 1 C(n).
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Erdős–Szemerédi 3-sunflower-free capacity
A family of three distinct sets A,B,C is a 3-sunflower (or Δ-system) if A∩ B = A∩ C = B∩ C. A family of sets is sunflower-free if it contains no 3-sunflower (equivalently, no sunflower of any size ≥ 3). Let [n]:=\1,2,…,n\ and let f(n) denote the maximum size of a sunflower-free family F⊆ 2^[n].
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Exponential growth constant for diagonal Ramsey numbers
C_17 is the limit (if it exists) of R(k)^1/k as k → ∞, where the diagonal Ramsey number R(k) is the smallest integer n such that every red/blue colouring of the edges of the complete graph K_n contains a monochromatic copy of K_k.
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Furstenberg–Sárközy exponent for square-difference-free sets
Let r(N) be the maximum size of a subset A⊂\1,…,N\ with no non-zero square differences a-b=n^2.Then C_4b is the least constant such that r(N) ≤ N^C_4b+o(1).
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Golay's Merit Factor
For n ≥ 1, let U_n denote the set of polynomials p(z) of degree n with coefficients ± 1.
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Good asymptotic constructions of Szemerédi–Trotter
If n,m are natural numbers, let C(n,m) denote the maximum number of incidences that are possible between n points and m lines in the plane. Establish upper and lower bounds on C(n,m) that are as strong as possible.
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Kakeya and Nikodym sets in finite fields
Let d ≥ 1, and let q be a prime power. Let 𝔽_q be a finite field of order q. A Kakeya set is a set K that contains a line in every direction, and an Nikodym set N is a set with the property that every point x in 𝔽_q^d is contained in a line that is contained in N ∪ x.
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Kakeya-type sum-difference constant
C_3b = SD(\0,1,∞\;-1) is the least exponent such that one has the inequality |A stackrelG- B| ≤ max(|A|, |B|, |A stackrelG+ B|)^C_3b whenever A, B are finite subsets of reals and G ⊂ A × B, where A stackrelG± B := a ± b: a ∈ A, b ∈ B.
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Komlós discrepancy constant
C_24 is the Komlós discrepancy constant (often denoted K). For a real matrix A∈ℝ^m× n, define its (sign) discrepancy by disc(A) := min_x∈-1,1^n ‖Ax‖_∞. For each n≥ 1, define the dimension-n Komlós discrepancy K_n := supdisc(A): A∈ℝ^n× n and ‖A_ast j‖_2≤ 1 for all columns j.
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Marton's conjecture (Polynomial Freiman-Ruzsa) constant
C_18 is the least constant such that, whenever A is a subset of 𝔽_2^n with lvert A+Arvert ≤ Klvert Arvert, then A can be covered by K^C_18+o(1) cosets of a subspace of cardinality at most lvert Arvert, where the limit o(1) is with respect to the limit K → ∞.
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Rate at which κ(n) approaches 1
Given a real matrix A, let its condition number be κ(A):=σ_max(A)/σ_min(A), where σ_min(A) and σ_max(A) denote the smallest and largest singular values of A, respectively (with κ(A)=∞ if σ_min(A)=0).
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Sidon set density inside (4,5) sets
C_5b is the largest constant such that every (4,5)-set of size n (i.e., a set of reals such that every four-element subset determines at least five distinct differences) contains a Sidon set of cardinality C_5bn.
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Single-set sum-difference exponent
For a finite nonempty subset A of an abelian group, write σ(A) := |A+A|/|A|, δ(A) := |A-A|/|A| for the doubling and difference constants. Ruzsa [Ru96] proved δ ≤ σ^2, and the Plünnecke–Ruzsa inequalities give the converse σ ≤ δ^2 [Bl26].
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Stanley–Wilf limit for the permutation pattern 1324
Let Av_n(1324) be the set of permutations of \1,2,…,n\ that avoid the permutation pattern 1324, and let S_n(1324) := |Av_n(1324)|. <a href="#CJS12-def-Sn">[CJS12-def-Sn]</a> The Stanley–Wilf limit (growth constant) for the pattern 1324 is C_30 := lim_n→∞ bigl(S_n(1324)bigr)^1/n.
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Subsets of the grid with no isosceles triangles
For n a natural number, let C(n) denote the size of the largest subset of [n]^2 = 1,…,n^2 that does not contain a (possibly flat) isosceles triangle. In other words, C(n) := max_S⊂ [n]^2|S|: a,b,c∈ S distinct implies ‖a-b‖ ≠ ‖b-c‖.
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Sum-product exponent for the reals
For a finite set A ⊂ ℝ write A+A = \ a+b : a,b ∈ A \, AA = \ ab : a,b ∈ A \ for the sumset and product set. The (real) sum-product exponent is C_84b := liminf_n → ∞ min_substackA ⊂ ℝ \ lvert Arvert = n log max(lvert A+Arvert, lvert AArvert)/log n.
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Sum-product problems
Given a natural number N and a ring R of size at least N, let C(R, N) denote the least possible value of max(|A+A|, |A · A|) where A ranges over subsets of R of cardinality N. Establish upper and lower bounds for C(R, N) that are as strong as possible.
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The Arithmetic Kakeya Conjecture
For each slope r ∈ ℝ ∪ ∞ define the projection π_r : ℝ^2 → ℝ by π_r(a,b) = a + rb for r ≠ ∞ and π_∞(a,b)=b.
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The Gyarmati-Hennecart-Ruzsa sum-difference constant
C_3a is the largest constant such that there exist arbitrarily large sets A,B of integers such that |A+B| ≪ |A| and |A-B| ≫ |A+B|^C_3a.
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The hypergraph Turán number of the tetrahedron
Let C be the largest quantity such that, as n → ∞, one can locate a 3-uniform hypergraph on n vertices and at least (C-o(1)) C(n, 3) edges that contains no copy of the tetrahedron K^(3)_4. What is C?
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Unnormalized single-set sum-difference exponent
For a finite non-empty set A of integers, C_3e is the least constant such that |A - A| ≤ |A + A|^C_3e for every such A; equivalently, C_3e = sup_A loglvert A-Arvert / loglvert A+Arvert. This is Problem 6.43 of [GGSWT2025].
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Babai–Seress Conjectures on the Diameter of Finite Groups
Babai–Seress Conjecture (Conjecture 1.5): There exists an absolute constant C such that the diameter of the alternating group A_n satisfies diam(A_n) ≤ n^C. Reference: L. Babai and Á. Seress, On the diameter of permutation groups, European Journal of Combinatorics 13 (1992), Conjecture 1.580029-0)
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Barker sequences
Every Barker sequence has length at most 13.
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Beaver Math Olympiad (BMO)
BMO#1) Let (a_n)_n ≥ 1 and (b_n)_n ≥ 1 be two sequences such that (a_1, b_1) = (1, 2) and (a_n+1, b_n+1) = begincases (a_n-b_n, 4b_n+2) & if a_n ≥ b_n cr (2a_n+1, b_n-a_n) & if a_n < b_n endcases for all positive integers n. Does there exist a positive integer i such that a_i = b_i?
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Beck–Fiala theorem and conjecture
The Beck–Fiala conjecture There exists a universal constant C > 0 such that every set system S_1, …, S_m ⊆ [n] of degree at most t admits a colouring χ : [n] → -1, +1 with |Σ_j ∈ S_i χ(j)| ≤ C √(t) for every i.
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Ben Green's Open Problem 1
Let A be a set of n positive integers. Does A contain a sum-free set of size at least frac n 3 + Ω(n), where Ω(n) → ∞ as n → ∞?
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Ben Green's Open Problem 16
What is the largest subset of [N] with no solution to x + 3y = 2z + 2w in distinct integers x, y, z, w?
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Ben Green's Open Problem 18
Suppose that G is a finite group, and let A ⊂ G × G be a subset of density α. Is it true that there are ≫_α |G|^3 triples x, y, g such that (x, y), (gx, y), (x, gy) all lie in A? Note: A is taken as α-dense, i.e. |A| ≥ α |G|^2 [Au16, Question 2]
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Ben Green's Open Problem 21
Suppose that a_1, …, a_k are integers which do not satisfy Rado's condition: thus if Σ_i ∈ I a_i = 0 then I = ∅. It then follows from Rado's theorem that the equation a_1x_1 + ⋯ + a_kx_k = 0 is not partition regular.
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Ben Green's Open Problem 31
Can we improve the lower bound N^1/2 + O(1), at least for infinitely many N?
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Ben Green's Open Problem 33
Are there infinitely many q for which there is a set A ⊂ ℤ/qℤ, |A| = (√(2) + o(1))q^1/2, with A + A = ℤ/qℤ? [Gr24]
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