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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Erdős Problem 423 [Er77c, p.71; ErGr80, p.83]: Let a(1) = 1, a(2) = 2, and for k ≥ 3 let a(k) be the least integer greater than a(k-1) that is a sum of at least two consecutive terms of the sequence. What is the asymptotic behaviour of this sequence? It seems likely that a_n = n + o(n).
Erdős Problem 44: Let N ≥ 1 and A ⊆ 1,…,N be a Sidon set. Is it true that, for any ε > 0, there exist M = M(ε) and B ⊆ N+1,…,M such that A ∪ B ⊆ 1,…,M is a Sidon set of size at least (1−ε)M^1/2?
Let A be a finite set and B= n ≥ 1 : a| ntextrm for some a∈ A. Is it true that, for every m>n≥ max(A), lvert B∩ [1,m]rvert /m< 2lvert B∩ [1,n]rvert/n?
For every x ∈ ℝ let A_x ⊂ ℝ be a bounded set with outer measure < 1. Must there exist an infinite independent set, that is, some infinite X ⊆ ℝ such that x ∉ A_y for all x ≠ y ∈ X? If the sets A_x are closed and have measure < 1, then must there exist an independent set of size 3?
Let r ≥ 3, and let f_r(N) denote the size of the largest subset of 1,…,N such that no subset of size r has the same pairwise greatest common divisor between all elements.
Let h(n) be maximal such that, for any set A⊆ ℕ of size n, the set a/(a,b): a,b∈ Ahas size at least h(n). Estimate h(n).
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.
Let F(n,α) denote the smallest m such that there exists a 2-colouring of the edges of K_n so that every X⊆ [n] with lvert Xrvert≥ m contains more than α C(lvert Xrvert, 2) many edges of each colour. Prove that, for every 0≤ α < 1/2, F(n,α)∼ c_αlog n for some constant c_α depending only on α.
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. Is there some constant c>0 such that R_3(n) ≥ 2^2^cn?
Erdős Problem 567 (Q3) Is Q_3 (the 3-dimensional hypercube) Ramsey size linear?
Erdős Problem 593 (\500): Characterize those finite 3-uniform hypergraphs which appear in every 3-uniform hypergraph of chromatic number > aleph_0. The answer is the set of obligatory finite 3-uniform hypergraphs, represented here on the labelled vertex sets Fin n.
Let r ≥ 2. Is it true that e(n,r+1) - e(n,r) → ∞ as n → ∞?
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).
Let r≥ 3. If the edges of K_r^2+1 are r-coloured then there exist r+1 vertices with at least one colour missing on the edges of the induced K_r+1. In other words, there is no balanced colouring. A conjecture of Erdős and Gyárfás [ErGy99].
Let X be a finite set of size n and H(n) be such that there is a function f:A : A⊆ X→ X so that for every Y⊆ X with lvert Yrvert ≥ H(n) we have f(A) : A⊆ Y=X. Prove that H(n)-log_2 n → ∞.
Let x_1,…,x_n∈ ℝ^2 and let R(x_i)=\# lvert x_j-x_irvert : j≠ i, where the points are ordered such that R(x_1)≤ ⋯ ≤ 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) ≥ (1-o(1))n?
Let F be a family of sets closed under taking subsets (i.e. if B⊆ AinF then B∈ F). There exists some element x such that whenever F'⊆ F is an intersecting subfamily we have lvert F'rvert ≤ lvert A∈ F : x∈ Arvert.
If there is a finite projective plane of order n then must n be a prime power?
What is the supremum of the set of admissible numbers?
Is every proportionately dissociated (infinite) set the union of a finite number of dissociated sets?
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).
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. Estimate f_c(n).
Is it true that R(n+1)/R(n)≥ 1+c for some constant c>0, for all large n?
Erdős Problem #817
F(n) / log n → ∞ as n → ∞
Does there exist a k>2 such that the k-sized subsets of 1,...,2k can be coloured with k+1 colours such that for every A⊂ 1,…,2k with lvert Arvert=k+1 all k+1 colours appear among the k-sized subsets of A?
Is it true that, for all large n, f(n + 1) ≥ f(n)?
Estimate m(n,k), or better give an asymptotic formula.
Erdős Problem 872, part (i) (weak form): there exists a constant ε > 0 such that the game length is at least ε · n for all sufficiently large n.
Let A ⊂ ℕ be an additive basis of order k which is minimal in the sense that if B ⊂ A is any infinite set, then A B is not a basis of order k. Must there exist an infinite B ⊂ A such that A B is an additive basis of order k + 1?
Does the limit lim_n→∞ f(2n)/f(n) tend to infinity? (Other finite limits have been ruled out by [KoLu25], see below)
Is the upper density of the set of odd numbers that cannot be expressed as a prime plus two powers of 2 positive?
Let S ⊆ ℝ be a set containing no solutions to a + b = c. Must there be a set A ⊆ ℝ ∖ S of cardinality continuum such that A + A ⊆ ℝ∖ S?
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.
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?
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.
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.
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?
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?
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)?
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)?
Can we improve the best upper bound? The base c must be positive, since =O compares norms.
If A ⊂ ℤ/pℤ is random, |A| = √(p), can we almost surely cover ℤ/pℤ with 100√(p) translates of A? [Gr24]
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.
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)?
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?
Problem 9 (ii): is r_5(N) ≪ N(log N)^-c?
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.
For N = 6 and all D ≥ 3, does there exist no solution to the monochromatic quantum graph equation system over ℂ?
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
