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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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).
For n ≥ 1, let U_n denote the set of polynomials p(z) of degree n with coefficients ± 1.
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.
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.
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‖.
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.
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.
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?
