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.
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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)
Every Barker sequence has length at most 13.
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?
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.
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 → ∞?
What is the largest subset of [N] with no solution to x + 3y = 2z + 2w in distinct integers x, y, z, w?
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]
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.
Can we improve the lower bound N^1/2 + O(1), at least for infinitely many N?
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]
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.
Does f(r) → ∞? [Gr24]
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.
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))?
Suppose A, B ⊆ 1, …, N both have size at least N^0.49. Must the sumset A + B contain a composite number?
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?
Given n points in the unit disc, must there be a triangle of area at most n^-2+o(1) determined by them?
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.
Conjecture 3.2 in [Wa2011]: Each Latin square of odd order has at least one transversal.
No closed-form expression that allows efficient computation of Dedekind numbers is currently known.
For n > 8, 2^n is not the the sum of distinct powers of 3. Expressed here in terms of the base 3 digits of n. This conjecture is equivalent to the halting of a 15-state 2-symbol Turing Machine. TODO(lezeau): Formalize the Turing Machine version of this problem.
Is there some k such that every large integer is the sum of a prime and at most k powers of 2?
Let f(n;r,k) be the maximal number of edges in an r-uniform hypergraph which contains no set of k many independent edges. For all r≥ 3, f(n;r,k)=max(C(rk-1, r), C(n, r)-C(n-k+1, r)). Note: the source states the formula with no range on n or k, but some restriction is needed: e.g.
Are there infinitely many binomial coefficients with deficiency 1?
Let f(N) be the size of the largest subset A⊆ 1,…,N such that every n∈ A+A is squarefree. Estimate f(N). In particular, is it true that f(N)≤ N^o(1), or even f(N) ≤ (log N)^O(1)? This theorem formalizes the subpolynomial bound as f(N) = O(N^ε) for every ε > 0.
Let p>q≥ 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. If p,q≠ 2,3 then what can be said about the density of non-representable numbers?
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.
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?
Erdős Problem 1167. Let r ≥ 2 be finite, γ ≥ 2, and λ be an infinite cardinal. Let κ_α > r be cardinals for all α < γ. Is it true that 2^λ → (κ_α + 1)_α < γ^r+1 implies λ → (κ_α)_α < γ^r? Here + means cardinal addition, so that κ_α + 1 = κ_α if κ_α is infinite. A problem of Erdős, Hajnal, and Rado.
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?
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].
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?
Does 1,2^3,…,N^3 contain a Sidon set of size ≫ N?
Let k≥3. Are there k consecutive primes in arithmetic progression?
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→ ∞?
Is it true that for every k ≥ 1 we have F(N + k) ≤ F(N) + 1 for all sufficiently large N?
Does there exist a maximal Sidon set A⊂ 1,…,N of size O(N^1/3)? A question of Erdős, Sárközy, and Sós [ESS94].
Let A be an infinite B₂[2] set. Must liminf |A ∩ 1, ..., N| * N ^ (- 1 / 2) = 0?
Estimate h(n) by finding a better upper bound.
The problem is to determine the limit of the sequence F(N)/√(N) as N → ∞.
Is it true that in any finite colouring of ℕ there exist arbitrarily large finite A such that all sums and products of distinct elements in A are the same colour?
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) ≪ 2^n.
What is the smallest k such that 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?
What is the largest k such that in any permutation of ℤ there must exist a monotone k-term arithmetic progression x_1 < ⋯ < x_k? Here a permutation of ℤ is a one-sided arrangement a_1, a_2, a_3, … of the integers, i.e.
Must every permutation of ℕ, contain a monotone 4-term arithmetic progression?
Can ℕ be partitioned into two sets, each of which can be permuted to avoid monotone 3-term arithmetic progressions?
Does the longest arithmetic progression of primes in 1,…,N have length o(log N)?
Is there an integer m with (m, 6) = 1 such that none of 2^k · 3^ℓ · m + 1 are prime, for any k, ℓ ≥ 0?
Let f(n) count the number of solutions to n=p+2^k for prime p and k≥ 0. Show that f(n)=o(log n).
Is it true that f(N)∼ N^1/3? Originally asked to Erdős by Bose. This is discussed in problem C11 of Guy's collection [Gu04].
