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.
10 shown
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.
Prove that there is always a prime between n^2 and (n+1)^2. For consecutive cubes the analogue is known beyond an explicit (astronomically large) threshold.
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.
Determine the growth of u(n), the maximum number of unit distances among n points in the plane. Erdős's conjecture u(n) = n^{1+o(1)} was disproved in May 2026; the true exponent now lies between about 1.014 (Sawin) and 4/3 (Spencer–Szemerédi–Trotter).
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.
Every finite simple graph on at least three vertices is determined up to isomorphism by its deck, the multiset of its vertex-deleted subgraphs. Verified by computer for all graphs up to 13 vertices; open in general.
Does every bounded linear operator on a separable infinite-dimensional complex Hilbert space have a non-trivial closed invariant subspace? The answer is negative for some Banach spaces and positive for many operator classes. The Hilbert space case is open.
Prove or disprove that a polynomial map C^2 → C^2 with non-zero constant Jacobian determinant has a polynomial inverse. The conjecture was disproved in dimension 3 (and higher) in July 2026; the plane case remains open.
Show that every Kakeya (Besicovitch) set in R^n has Hausdorff and Minkowski dimension n. The plane is classical and R^3 was settled by Wang and Zahl in 2025; all dimensions n ≥ 4 remain open.
Prove that there are infinitely many primes p with p + 2 prime. Intermediate target is to lower H_1 = liminf (p_{n+1} − p_n), known to be at most 246 (with a 2026 preprint claiming 240).
