Recent progress on open problems
New bounds, settled cases and verified computations on the problems we track, since 2020, newest first. Each entry links to the paper or record it comes from; the problem pages put it in context. For where each problem stands today, see the status table.
2026
- Approximation ratio and integrality gap for metric TSP gap ≤ 4/3 verified for n ≤ 15 — Cook, Hougardy, Petrich · source · Algorithms
- Bounds on the de Bruijn–Newman constant Λ Λ ≤ 0.1787854 (unrefereed) — Gomila, with Romik and Atkin · source · Analysis
- Erdős minimum overlap problem M ≤ 0.380868 — SimpleTES · source · Combinatorics
- Frankl's union-closed sets conjecture ≈ 0.38289 (preprint) — Costa & Sadhu · source · Combinatorics
- Hadamard matrices of open orders all 12 open orders below 2000 constructed (announced) — Alpöge, Voinov, Reynolds-Haertle, with Claude · source · Combinatorics
- Kissing configurations in dimensions 10–31 τ₁₉ ≥ 11,948 — B. S. Ho · source · Geometry
- Kissing configurations in dimensions 10–31 τ₁₁ ≥ 604 — Bianchi, Kwon, Pappu, Zou · source · Geometry
- Kissing configurations in dimensions 10–31 τ₁₂ ≥ 841 — Takhanov, Yun, Assylbekov · source · Geometry
- Snake-in-the-box — longest induced paths in hypercubes n = 9–13: 191, 379, 746, 1476, 2924 — Orland, Fagan et al. · source · Combinatorics
- Tensor rank of 3×3 matrix multiplication rank ≥ 20 over F₂ — C. Wang · source · Algorithms
- Tensor rank of 3×3 matrix multiplication rank ≥ 21 over F₂ (preprints) — C. Wang; S. Yang, R. Zhao et al. · source · Algorithms
- The 1/3–2/3 conjecture for balanced pairs in posets verified for all posets with ≤ 14 elements (preprint) — A. Gupta · source · Combinatorics
- The 1/3–2/3 conjecture for balanced pairs in posets δ ≥ (5−√5)/10 + ε (preprint) — Aires, Chan, Pak, Panova · source · Combinatorics
- The Erdős unit distance problem in the plane α > 1 (conjecture disproved) — OpenAI model; Alon, Bloom, Gowers et al. · source · Geometry
- The Erdős unit distance problem in the plane α ≥ 1.014 — Sawin · source · Geometry
- The exponent ω of matrix multiplication ω < 2.371177 (preprint) — Dupont et al. · source · Algorithms
- The lonely runner conjecture 11–13 runners — Sungkawichai & Trakulthongchai · source · Combinatorics
- The lonely runner conjecture 14 and 15 runners (claimed preprint) — Allikvee · source · Combinatorics
- The Riemann Hypothesis > 2/3, simple and on the line (unrefereed) — Anthropic research model; Lamzouri · source · Number theory
- The Thomson problem (minimum-energy charges on a sphere) n = 7, numerically certified — Kuznetsov & Sahinidis · source · Optimisation
- The Thomson problem (minimum-energy charges on a sphere) n = 8 (Lean-verified preprint) — Kryvonos, Liehr, Taylor · source · Optimisation
- The twin prime conjecture and bounded prime gaps H₁ ≤ 240 (preprint) — J. Stadlmann · source · Number theory
- The twin prime conjecture and bounded prime gaps H₁ ≤ 186 (unrefereed) — OpenAI · source · Number theory
2025
- Deciding hard small Turing machines (Busy Beaver) Σ(6) > 10↑↑10↑↑10↑↑8 — mxdys · source · Computability
- Erdős minimum overlap problem M ≤ 0.380924 — AlphaEvolve · source · Combinatorics
- Kissing configurations in dimensions 10–31 τ₁₁ ≥ 593 — AlphaEvolve · source · Geometry
- Large cap sets in F_3^n capacity ≥ 2.2203 — Zhai et al. (X-evolve) · source · Combinatorics
- Precision critical exponents of the 3D Ising universality class Δσ = 0.518148806(24), Δε = 1.41262528(29) — Chang et al. · source · Theoretical physics
- Tensor rank of 4×4 matrix multiplication 48 over Q — Dumas, Pernet, Sedoglavic · source · Algorithms
- The Collatz (3n + 1) conjecture verified for all n < 2^71 — D. Barina · source · Number theory
- The Erdős–Straus conjecture verified for n ≤ 10^18 — Mihnea & Dumitru · source · Number theory
- The Kakeya conjecture in dimensions n ≥ 4 Kakeya sets in R³ have dimension 3 — Wang & Zahl · source · Analysis
- The lonely runner conjecture 8 runners — Rosenfeld · source · Combinatorics
- The lonely runner conjecture 9 and 10 runners — Trakulthongchai; Rosenfeld (9) · source · Combinatorics
2024
- Deciding hard small Turing machines (Busy Beaver) BB(5) = 47,176,870 (Rocq proof) — bbchallenge collaboration · source · Computability
- Legendre's conjecture cubes: n ≥ exp(exp(32.76)) — Mossinghoff, Trudgian, Yang · source · Number theory
- The exponent ω of matrix multiplication ω < 2.371339 — Alman, Duan, Vassilevska Williams, Xu, Xu, Zhou · source · Algorithms
- The Ramsey number R(4,6) R(4,6) ≤ 40 — Angeltveit & McKay · source · Graph theory
- The Ramsey number R(5,5) R(5,5) ≤ 46 — Angeltveit & McKay · source · Graph theory
2023
- Cap sets in dimension 8 r₃(8) ≥ 512 — Romera-Paredes et al. (FunSearch) · source · Combinatorics
- Frankl's union-closed sets conjecture ≈ 0.38271 — Liu · source · Combinatorics
- Large cap sets in F_3^n capacity ≥ 2.2202; r₃(8) ≥ 512 — Romera-Paredes et al. (FunSearch) · source · Combinatorics
- The log-rank conjecture in communication complexity D ≤ O(√r) — Sudakov & Tomon · source · Complexity
2022
- Deciding hard small Turing machines (Busy Beaver) S(6) > 10↑↑15 — S. Ligocki & P. Kropitz · source · Computability
- Erdős minimum overlap problem M ≥ 0.379005 — White · source · Combinatorics
- Explicit Boolean circuit lower bounds 3.1n − o(n) — Li & Yang · source · Complexity
- Frankl's union-closed sets conjecture constant 0.01 — Gilmer · source · Combinatorics
- Frankl's union-closed sets conjecture ≈ 0.38197 — Alweiss–Huang–Sellke; Chase–Lovett; Sawin; Pebody · source · Combinatorics
- Optimal Golomb rulers OGR-28 optimal (585) — distributed.net · source · Combinatorics
- Tensor rank of 4×4 matrix multiplication 47 over GF(2) — Fawzi et al. (AlphaTensor) · source · Algorithms
- The exponent ω of matrix multiplication ω < 2.371866 — Duan, Wu, Zhou · source · Algorithms
2021
- Crossing numbers of complete and complete bipartite graphs cr(K₁₃) = 225, cr(K₁₄) = 315 — Aichholzer et al. · source · Graph theory
- Legendre's conjecture cubes: n ≥ exp(exp(32.892)) — Cully-Hugill · source · Number theory
- Packing equal circles in a unit square optimality proven for n = 31, 32, 33 — M. C. Markót · source · Optimisation
2020
- Approximation ratio and integrality gap for metric TSP ratio 3/2 − 10⁻³⁶ — Karlin, Klein, Oveis Gharan · source · Algorithms
- Bounds on the de Bruijn–Newman constant Λ Λ ≤ 0.2 — Platt & Trudgian · source · Analysis
- Smallest sorting networks for 13+ inputs n = 11: 35 and n = 12: 39 proven optimal — Harder · source · Algorithms
- The chromatic number of the plane (Hadwiger–Nelson problem) 509 vertices — Parts (Polymath16) · source · Combinatorics
- The Riemann Hypothesis > 5/12 — Pratt, Robles, Zaharescu, Zeindler · source · Number theory
Something missing? Results published elsewhere can be reported on a problem's page as a literature claim; they are checked before they count. Open problems an AI can work on · Guides