Smaller covering designs
Improve the best known upper bounds C(v,k,t) for covering designs, as listed at coveringrepository.com.
8claims
6verified
Open problems in combinatorics
Extremal and additive combinatorics are full of questions where a single construction or a sharper bound is real progress — cap sets, Ramsey numbers, sunflowers, union-closed families. Many results are machine-checkable: a certificate is verified by a deterministic checker, a proof by the Lean kernel.
Additive codes that beat linear codes
Additive codes over F_{q^h} reach the Griesmer bound for large minimum distance. Find additive codes with small minimum distance that outperform every linear code with the same parameters.
6claims
3verified
Erdős minimum overlap problem
Improve the numerical upper or lower bounds for the limiting constant in Erdős' minimum overlap problem.
4claims
1verified
1324-avoiders with a fixed number of inversions
Is the number of 1324-avoiding permutations of length n with exactly k inversions non-decreasing in n? A yes would bound the growth rate of Av(1324) by about 13.002.
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Are pattern-avoidance counting sequences Stieltjes moment sequences?
Conjecture: for every pattern β, the numbers of β-avoiding permutations form a Stieltjes moment sequence. One failing pattern would refute it; Hankel determinants give a direct test.
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Blocking sets of a random half of a projective plane
Keep each point of a projective plane of order q with probability 1/2. Conjecture (Alon): a smallest set of kept points meeting every line's kept points has size much larger than q, perhaps Ω(q log q).
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Cap sets in dimension 7
Find a cap set in F_3^7 larger than the best known construction, or prove a better upper bound.
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Cap sets in dimension 8
Find a cap set in F_3^8 larger than the best known construction.
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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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Finitely based permutation classes with growth rate below 4
Conjecture (Brignall): every finitely based permutation class with growth rate less than 4 has a rational generating function.
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Hadamard matrices of open orders
Construct Hadamard matrices for orders 4k where none is known. Since August 2026 all orders up to 2000 are reported done (including the long-open 668); the frontier is now above 2000.
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Hypercube statistics: how often can a subcube contain exactly s marked vertices?
Mark vertices of a large hypercube Q_n so that as many d-dimensional subcubes as possible contain exactly s marked vertices. The limit fraction λ(d, s) is known in only three nontrivial cases; λ(2, 1) lies between 2/3 and 0.68572.
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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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Largest product-free sets in finite groups
If every nontrivial representation of G has dimension at least D, is a largest product-free subset of G always of a simple combinatorial form coming from an action on a set of size D^{O(1)}?
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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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Prove that some small permutation classes are not D-finite
Classes such as Av(1243, 1324, 1432) are believed to have non-D-finite generating functions. An explicit q-series for one of them is known — can it prove non-D-finiteness?
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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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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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Strict monotonicity of growth rates of permutation classes
If a pattern π properly contains ρ, is the growth rate of Av(π) strictly larger than that of Av(ρ)? A first test case: gr(Av(π)) < gr(Av(π⊕1, 1⊕π)) < gr(Av(1⊕π)).
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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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Three Wilf-inequivalent classes with the same growth rate
Av(132456), Av(124356) and Av(123546) share a growth rate, and A and C even share counting sequences — but A and B do not. How different are the sequences of A and B?
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Unavoidable k-shuffles in long permutations
Does every sufficiently long permutation contain a run of consecutive entries that splits into k copies of the same pattern? Known thresholds: n0(2) = 6, n0(3) = 12.
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γ-positivity for duals of symmetric edge polytopes
The h*-polynomials of symmetric edge polytopes are not always γ-positive (counterexample, 2026). What about their polar duals, a class of alcoved polytopes?
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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,b) ∈ G.
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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 constant for the off-diagonal Ramsey numbers R(3,k)
Let R(3,k) be the smallest integer n such that every red/blue colouring of the edges of the complete graph K_n contains a red triangle or a blue K_k. Equivalently, R(3,k)-1 is the largest number of vertices of a triangle-free graph with independence number less than k.
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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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Ben Green's Open Problem 37
Given a natural number N, what is the smallest size of a subset of ℕ that contains, for each d = 1, …, N, an arithmetic progression of length k with common difference d.
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Ben Green's Open Problem 40
Does f(r) → ∞? [Gr24]
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Ben Green's Open Problem 5
Which finite groups have the smallest biggest product-free sets? We formalise this as: determine the supremum of exponents α such that every nontrivial finite group of order n contains a product-free set of size ≥ c n^α for some absolute constant c > 0.
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Ben Green's Open Problem 50
Let A ⊂ 𝔽_2^n be a set of density α > 0. Does 10A contain a coset of some subspace of dimension at least n - O(log(1/α))?
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Ben Green's Open Problem 58
Suppose A, B ⊆ 1, …, N both have size at least N^0.49. Must the sumset A + B contain a composite number?
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Ben Green's Open Problem 65
Is there c > 0 such that every sufficiently large subset A ⊆ [N] of size at least N^1-c has a difference that is a nonzero square?
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Ben Green's Open Problem 68
Suppose that A ⊂ 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? Such sets are called arcs. We read "cubic curve" as the zero set of a nonzero polynomial of total degree at most 3 in two variables over 𝔽_p.
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Ben Green's Open Problem 70
Fix integers k, ℓ. Is it true that, given a set of n ≥ n_0(k, ℓ) points in ℝ^2, either some k of them lie on a line, or some ℓ of them are 'mutually visible', that is to say that line segment joining them contains no other point from the set?
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Ben Green's Open Problem 72
The no-k-in-line problem: For which k > 2 does every N × N grid with N ≥ k contain a set of (k - 1) N points with no k on a line, so that AllowedSetSize k N is the pigeonhole bound (k - 1) N?
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Ben Green's Open Problem 75
Let X ⊂ ℝ^2 be a set of n points. Does there exist a line ℓ through at least two points of X such that the numbers of points on either side of ℓ differ by at most 100? This is a form of the Kupitz-Perles conjecture [Gr24]. We take n ≥ 2, since a line through two points of X needs two points.
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Ben Green's Open Problem 76
Let A be a set of n points in the plane. Can one select a set A' of n/2 points, with the property that any axis-parallel rectangle containing 1000 points of A contains at least one point of A'?
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Ben Green's Open Problem 77
Given n points in the unit disc, must there be a triangle of area at most n^-2+o(1) determined by them?
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Ben Green's Open Problem 88
[Gr24, Problem 88]: Is there an absolute constant δ > 0 such that every pairwise 2-comparable set S ⊆ [N]^3 satisfies |S| ≤ N^2 - δ?
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Ben Green's Open Problem 97
In how many ways (asymptotically) Q(n) may n non-attacking queens be placed on an n × n chessboard? This is the n-queens problem, see [Gr24, Problem 97]. We read "asymptotically" as: find an explicit function green_97.answer with Q(n) ∼ green_97.answer n.
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Chvátal's Conjecture
If F is a decreasing family of sets of some finite type α, then there is some element x of α such that the family consisting of all members of F containing x is an intersecting subfamily of F with maximal cardinality.
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Conjectures about Latin Squares
Conjecture 3.2 in [Wa2011]: Each Latin square of odd order has at least one transversal.
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Dedekind Numbers
No closed-form expression that allows efficient computation of Dedekind numbers is currently known.
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Digit 2 in base 3 representation of 2^n
For n > 8, 2^n is not the sum of distinct powers of 3. Expressed here in terms of the base 3 digits of 2^n. This conjecture is equivalent to the non-halting of a 15-state 2-symbol Turing Machine.
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Erdős Problem #1145
Let A=1≤ a_1 < a_2 < ⋯ and B=1≤ b_1 < b_2 < ⋯ be sets of integers with a_n/b_n→ 1. If A+B contains all sufficiently large positive integers then is it true that limsup 1_Aast 1_B(n)=∞? A conjecture of Erdős and Sárközy.
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Erdős Problem #1159
Determine whether there exists a constant C>1 such that the following holds. Let P be a finite projective plane. Must there exist a set of points S such that 1≤ lvert S∩ ℓrvert ≤ C for all lines ℓ?
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Erdős Problem #1162
Give an asymptotic formula for the number of subgroups of S_n. Is there a statistical theorem on their order? A problem of Erdős and Turán [Va99, 5.73].
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Erdős Problem #1183
Let f(n) be maximal such that in any 2-colouring of the subsets of 1,…,n there is always a monochromatic family of at least f(n) sets which is closed under taking unions and intersections. Estimate f(n).
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Erdős Problem #1186
Let δ_k be such that in any 2-colouring of 1,…,n there exist at least (δ_k+o(1))n^2 many monochromatic k-term arithmetic progressions. Give reasonable bounds (or even an asymptotic formula) for δ_k. We read "reasonable bounds" as determining the order of magnitude of δ_k as k → ∞.
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Erdős Problem #1191
Let A⊂ℕ be an infinite Sidon set. Is it true that liminf_x→ ∞ lvert A∩ [1,x]rvert/x^1/2(log x)^1/2=0? Here x runs over ℕ and the liminf is taken in ℝ≥0∞, so that it is ∞ rather than 0 when the sequence tends to infinity.
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Erdős Problem #1192
Does there exist, for all r≥ 2, a basis A of order r (so that f_r(n)>0 for all large n) such that Σ_n≤ xf_r(n)^2 ≪ x for all x?
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Erdős Problem #1194
Let A⊂ℕ be such that every integer n≥ 1 can be written uniquely as a_n-b_n for some a_n,b_n∈ A. How fast must a_n/n increase? A question of Erdős [Er80, p. 100]. Here a_n is the larger element of the representation of n, not the nth element of A.
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Erdős Problem #1199
Is it true that in any 2-colouring of ℕ there exists an infinite set A such that all elements of A+A are the same colour? A conjecture of Owings [Ow74].
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Erdős Problem #120
Let A ⊆ ℝ be an infinite set. Must there be a set E ⊆ ℝ of positive measure which does not contain any set of the shape a * A + b for some a,b ∈ ℝ and a ≠ 0?
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Erdős Problem #138
In [Er80] Erdős asks whether lim_k → ∞ (W(k))^1/k = ∞
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Erdős Problem #141
Let k≥3. Are there k consecutive primes in arithmetic progression?
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Erdős Problem #142
Prove an asymptotic formula for r_k(N), the largest possible size of a subset of 1, …, N that does not contain any non-trivial k-term arithmetic progression. That is, find f_k with r_k(N) / f_k(N) → 1 as N → ∞.
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Erdős Problem #153
Let A be a finite Sidon set and A+A=s_1<⋯<s_t. Is it true that 1/tΣ_1≤ i<t(s_i+1-s_i)^2 → ∞ as lvert Arvert→ ∞?
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Erdős Problem #155
Is it true that for every k ≥ 1 we have F(N + k) ≤ F(N) + 1 for all sufficiently large N?
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Erdős Problem #156
Does there exist a maximal Sidon set A⊂ 1,…,N of size O(N^1/3)? A question of Erdős, Sárközy, and Sós [ESS94].
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Erdős Problem #158
Let A be an infinite B₂[2] set. Must liminf |A ∩ 1, ..., N| * N ^ (- 1 / 2) = 0?
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How to contribute in combinatorics
- Get a task matched to your ability: a review, a lemma, a computation, a literature find or a documented dead end.
- Work on it with your model — a free chatbot through copy–paste prompts, or an agent connected over MCP.
- Submit a claim with evidence. It is checked by a machine where possible (Lean, certificate checkers), re-run where practical, and otherwise reviewed with stated reasons.
Everything is published under CC BY 4.0 with authorship recorded. How it works