Erdős Problem #723
If there is a finite projective plane of order n then must n be a prime power?
No claims yet Be the first →
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.
Erdős Problem #749
Let ε>0. Does there exist A⊆ ℕ such that the lower density of A+A is at least 1-ε and yet 1_Aast 1_A(n) ≪_ε 1 for all n?
No claims yet Be the first →
Erdős Problem #789
Let h(n) be maximal such that if A⊆ ℤ with lvert Arvert=n then there is B⊆ A with lvert Brvert ≥ h(n) such that if a_1+⋯+a_r=b_1+⋯+b_s with a_i,b_i∈ B then r=s. Estimate h(n).
No claims yet Be the first →
Erdős Problem #817
Erdős Problem #817
No claims yet Be the first →
Erdős Problem #857
Estimate m(n,k), or better give an asymptotic formula.
No claims yet Be the first →
Green's Open Problem 12
Let G be an abelian group of size N, and suppose that A ⊂ G has density α. Are there at least α^15 N^10 tuples (x_1, …, x_5, y_1, …, y_5) ∈ G^10 such that x_i + y_j ∈ A whenever j ∈ i, i+1, i+2? Note: We interpret indices modulo 5.
No claims yet Be the first →
Green's Open Problem 15
Does there exist a Lipschitz function f : ℕ → ℤ whose graph Γ = (n, f(n)) : n ∈ ℕ ⊆ ℤ^2 is free of 3-term progressions?
No claims yet Be the first →
Green's Open Problem 22
If 1, …, N is r-coloured then, for N geqslant N_0(r), there are integers x, y geqslant 3 such that x + y, xy have the same colour. Find reasonable bounds for N_0(r). The goal is to improve upon the Green-Sawhney bound.
No claims yet Be the first →
Green's Open Problem 24
If A is a set of n integers, what is the maximum number of affine translates of the set lbrace 0,1,3 rbrace that A can contain? Conjectured in [Aa19] p.579: (1/3 + o(1)) n^2.
No claims yet Be the first →
Green's Open Problem 25
For which values of k is the following true: whenever we partition [N] = A_1 ∪ … ∪ A_k, |bigcup^k_i=1 (A_i hat+ A_i)| ≥ 1/10 N?
No claims yet Be the first →
Green's Open Problem 27
What is the size of the smallest set A ⊂ ℤ / pℤ (with at least two elements) for which no element in the sumset A + A has a unique representation?
No claims yet Be the first →
Green's Open Problem 32
Let p be a prime and let A ⊂ ℤ/pℤ be a set of size ⌊ √(p) ⌋. Is there a dilate of A containing a gap of length 100√(p)?
No claims yet Be the first →
Green's Open Problem 36
Do the following exist, for arbitrarily large n? An abelian group H with |H| = n^2+o(1), together with subsets A_1, ..., A_n, B_1, ..., B_n satisfying |A_i||B_i| ≥ n^2-o(1) and |A_i + B_i| = |A_i||B_i|, such that the sets A_i + B_i are disjoint from the sets A_j + B_k (j ≠ k)?
No claims yet Be the first →
Green's Open Problem 38
Can we improve the best upper bound? The base c must be positive, since =O compares norms.
No claims yet Be the first →
Green's Open Problem 39
If A ⊂ ℤ/pℤ is random, |A| = √(p), can we almost surely cover ℤ/pℤ with 100√(p) translates of A? [Gr24]
No claims yet Be the first →
Green's Open Problem 51
Suppose that A ⊂ 𝔽_2^n is a set of density α. What is the largest size of coset guaranteed to be contained in 2A? We phrase this by asking for the exact function F(α, n) giving the maximum dimension of a guaranteed coset.
No claims yet Be the first →
Green's Open Problem 52
Suppose that A ⊂ 𝔽_2^n is a set with an additive complement of size K. Does 2A contain a coset of codimension O_K(1)?
No claims yet Be the first →
Green's Open Problem 53
Suppose that 𝔽_2^n is partitioned in to sets A_1, ..., A_K. Does 2A_i contain a coset of codimension O_K(1) for some i?
No claims yet Be the first →
Green's Open Problem 9
Problem 9 (ii): is r_5(N) ≪ N(log N)^-c?
No claims yet Be the first →
Main conjecture on fusible numbers
If x is a fusible number and y is its successor, then the interval [x + 1, y + 1) can be divided into intervals [ℓₙ, ℓₙ₊₁), such that the fusible numbers in [ℓₙ, ℓₙ₊₁) are obtained by fusing the n + 1st successor of x with a fusible number.
No claims yet Be the first →
Monochromatic quantum graphs (inherited vertex colorings)
For N = 6 and all D ≥ 3, does there exist no solution to the monochromatic quantum graph equation system over ℂ?
No claims yet Be the first →
Number of different products of subsets of 1, 2, …, n
Conjecture: let p ≤ n be prime. If m and p^a m are two such products, then so is p^k m for all 0 < k < a. - Yan Sheng Ang, Feb 13 2020
No claims yet Be the first →
Ramsey numbers
The open problem: determine the Ramsey number R(5,5). It is known that 43 ≤ R(5,5) ≤ 46.
No claims yet Be the first →
Sparse Ruler
Wichmann's conjecture on optimal rulers. Every optimal ruler of sufficiently large length is a Wichmann ruler W(r, s) (up to reflection, i.e. reversing the segment list). Posed by Wichmann [Wi63].
No claims yet Be the first →
Steiner Systems
Construct an S(t, k, n)-Steiner system with n > k > t > 5, t < 10, and n < 200. No example of a Steiner system with t > 5 is known, despite a 2014 existence theorem by Keevash showing that such systems must exist for sufficiently large n. Reference: Large Steiner Systems
No claims yet Be the first →
The length of an s-increasing sequence of r-tuples
F(n) ≤ n^3/2.
No claims yet Be the first →
VCₙ dimension of convex sets in ℝⁿ, ℝⁿ⁺¹, ℝⁿ⁺²
Every convex set in ℝ^3 has VC_2 dimension at most 2.
No claims yet Be the first →
Written on the Wall II - Conjecture 40
WOWII Conjecture 40 For a nontrivial connected graph G the size f(G) of a largest induced forest satisfies f(G) ≥ ceil((p(G) + b(G) + 1)/2) where p(G) is the path cover number and b(G) is the largest induced bipartite subgraph size.
No claims yet Be the first →
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