Patterns, colorings, and discrete constructions, including Ramsey theory, cap sets, and geometric combinatorics.
14 records · Includes focused interests · How we rate progress
Math, Inc. reports Lean formalizations of sphere-packing optimality for the E₈ lattice in dimension eight and the Leech lattice in dimension 24, with uniqueness among periodic packings in dimension 24. The human maintainers report checking definitions, axiom dependencies and the supplied code.
Read original source Math, Inc.: reported formalization and scopemath.inc
Original report: 2026-03-02The manuscript reports arbitrarily large finite simple graphs with independence number at most two and chromatic number greater than their largest clique-minor order. It also claims violation of the ordinary fractional-coloring weakening.
Original report: 2026-10-06Ellenberg, Libedinsky, Plaza, Simental and Williamson prove that for n=2^m, dyadically well-distributed permutations form a Bruhat interval isomorphic to a hypercube of dimension m·2^(m−1). This has order n log n, within a constant factor of the maximum.
Read original source Original paper, Theorem 1.1 and methodologyarxiv.org
Original report: 2026-01-03OpenAI reports that every positive-integer set with divergent reciprocal sum contains arithmetic progressions of every finite length, through fixed-length bounds r_k(N)≤C_k N exp(−c_k(log N)^ε_k).
Read original source Original manuscript: claim, hypotheses, and contextgithub.com
Original report: 2026-10-06OpenAI reports a bipartite graph with 35 vertices and 66 edges whose homomorphism density in a finite simple host is below the predicted edge-density bound; it also reports failure of the associated forcing assertion.
Read original source Original manuscript: claim, hypotheses, and contextgithub.com
Original report: 2026-10-06OpenAI reports that every nonempty finite oriented graph has a vertex with at least as many vertices at directed distance exactly two as at directed distance one.
Read original source Original manuscript: claim, hypotheses, and contextgithub.com
Original report: 2026-10-06The artifact reports exact optimal radii and classifications of optimal packings for one through seven freely rotated unit squares in a disk.
Read original source Original researchgithub.com
Original report: 2026-10-02The human-verified writeup gives a fixed ε>0 and growing planar point sets with at least n^(1+ε) unit distances.
Original report: 2026-05-20The manuscript reports a nine-dimensional counterexample using rank-one orthogonal projectors on R⁴ with the Frobenius metric.
Read original source Original manuscript · family 156github.com
Original report: 2026-10-06The manuscript reports that five colors cannot suffice, implying that the plane’s chromatic number is six or seven if the result holds.
Read original source Original manuscript · family 158github.com
Original report: 2026-10-06The manuscript reports R(5,t) = t⁴/(log t)^(3+o(1)) as t tends to infinity.
Read original source Original manuscript · family 170github.com
Original report: 2026-10-06The authors report configurations packing 11 and 12 unit regular hexagons inside regular hexagons of approximate side lengths 3.931 and 3.942, respectively.
Read original source Georgiev, Gómez-Serrano, Tao, and Wagner paperarxiv.org
Original report: 2025-11-03The authors report new infinite families attaining R(B(n−1), B(n)) = 4n−1 for specified parameters, where B(k) is k triangles sharing an edge.
Original report: 2026-08-24The FunSearch paper reports new large cap-set constructions improving the authors’ stated finite-dimensional and asymptotic baselines.
Original report: 2023-12-14Patterns, colorings, and discrete constructions, including Ramsey theory, cap sets, and geometric combinatorics.
Related focused topics and problems retain their own statements and baselines. A contribution may belong to more than one field; it is counted once within each topic.
Background review: 2026-10-11. Selected sources:
Publication dates and editorial changes.
Gauss: AI-assisted sphere-packing formalization in dimensions 8 and 24
OpenAI reports: Erdős’s reciprocal-sum progression conjecture
OpenAI reports: Borsuk’s covering question in dimension nine
AlphaEvolve: improved hexagon-packing configurations reported
OpenAI reports: Borsuk’s covering question in dimension nine
Station: new infinite families for book Ramsey numbers reported
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