Problems

The classical Abbott-Hanson recurrence gives S(k+2) ≥ 9S(k)+4 for Schur numbers, and had stood as the basis for the best asymptotic lower bounds. Shifted S-templates, a more flexible form of Rowley's template construction, yield S(k+2) ≥…

Erdős, Herzog and Piranian (1958) asked whether the regular n-gon maximizes the product of pairwise distances among n points of fixed diameter. After the recent discovery that it does not for even n, this paper proves the first exponential…

For A=0,1,2^d, write f(d) for the fewest proper sub-boxes covering every point exactly twice. Leader, Miličević and Tan asked whether f(d)≥ 2^d for all d, as Question 4.1 of the PatternBoost paper. The paper gives new bounds on f(d).

Seymour conjectured that every finite oriented graph has a vertex with at least as many exact second outneighbors as outneighbors. Known cases include tournaments (Fisher 1996) and minimum outdegree at most six (Kaneko-Locke 2001), and for…

What is the minimum asymptotic density δ_k of monochromatic k-term arithmetic progressions in every two-colouring of 1, …, n? The exact certificate gives δ_3 = 117/2192, matching the known 548-bead colouring.

Erdos and Hajnal asked whether h_r(G) = maxχ(H) : H ⊆ G, girth(H) ≥ r tends to infinity as χ(G) does, for every fixed r ≥ 4. It does in every fixed polynomial edge-density regime.

Kotzig conjectured that for every even n ≥ 4 the complete graph K_n decomposes into n-1 perfect matchings such that every pair of them forms a Hamilton cycle. An asymptotic version holds: K_n decomposes into n-1 perfect matchings of which…

R(c) = 40c+41 for every c ≥ 2 such that c+1 is divisible by 3, 4, 5, or 7 (covering ≈ 66% of all c); the full conjecture (Myers 2015 Conj. 4.9, ABEMRS16 §5.5) reduces to prime cases p ≥ 89, all smaller primes settled by SAT. Twenty-eight…

Espuny Diaz, Lichev and Wesolek conjectured that a Dirac-type minimum degree condition forces Hamiltonicity in spanning subgraphs of cycle powers. Asymptotically true: for every ε > 0 and all large k, any spanning subgraph of the kth power…

Improved lower bounds for nine classical Ramsey numbers, including R(3,13) ≥ 61, R(3,18) ≥ 100, and seven R(4,k) records up to R(4,20) ≥ 237, found by AlphaEvolve-discovered search algorithms.

How large can a Bruhat interval in S_n that is a poset hypercube be? Using a permutation pattern suggested by AlphaEvolve, the authors exhibit hypercube intervals of dimension O(n log n) for n a power of 2, matching the largest possible…

Tuza conjectured that every finite simple graph satisfies τ(G) ≤ 2ν(G), where ν counts pairwise edge-disjoint triangles and τ is the fewest edges whose deletion leaves the graph triangle-free. Puleo had proved it for maximum average degree…

Seymour conjectured that every oriented graph has a vertex x with |N^++(x)| ≥ |N^+(x)|. It holds for oriented graphs of minimum out-degree exactly 7, the first improvement to the out-degree threshold since Kaneko and Locke settled degree 6…

Pach conjectured that n Jordan arcs, pairwise crossing exactly once with no triple points, have O(n) tangent pairs. The best known bound stood at O(n^7/4); the paper improves it to O(n^3/2) (and O(n^5/3) in the at-most-one-crossing…

Deng, Tidor and Zhao asked whether [N] admits a coloring with N^o(1) colors and no symmetrically coloured 4-term arithmetic progression, giving an O(N^log_223) coloring. The paper gives an O_k(N^4/k^2) coloring of [N] avoiding…

Reiner conjectured a description of the homotopy types of intervals in higher Bruhat orders. In corank 3 it holds: the facial intervals of B(n,n-3) are exactly the spherical intervals, and every other interval is contractible.

Let A(k) be the largest possible number of moves in a north-east lattice path whose visited vertices contain no k collinear points. Gerver (1979) and Gerver and Ramsey (1979) bounded A(k) by exp(Ω(log(k)^2)) ≤ A(k) ≤ exp(O(k^4)), and…

Korsky, Saffat and Aiylam bounded the growth constant c(G) for integer-valued Lipschitz functions on G(n,d/n) between 1/(2d) and 4log^2 d/d up to lower-order terms. The random-graph side is sharpened.

Is the fractional chromatic number of every d-degenerate triangle-free graph at most (1+o(1))d/log d, with a matching lower bound, as conjectured by Martinsson and Steiner? The upper bound is confirmed constructively for graphs of girth at…

Let g(r) be the fewest edges in an r-uniform intersecting hypergraph with cover number r. Erdos and Lovasz proved g(r) ≥ 8r/3 - 3. An elementary argument gives g(r) ≥ 3r - 4, and building on it with Kahn's small-codegree edge-colouring…