Ramsey theory

Combinatorics5

• 160Sep 23, 2026 Explained

  Superexponential van der Waerden numbers

  Resolves Erdős's superexponential-growth question for van der Waerden numbers. If $W_r(k)$ is the least interval length forcing a monochromatic k-term progression in every r-coloring, then $W_r(k)\gt k^{ck\lfloor\log_2 r\rfloor}$ for an absolute c > 0, all r ≥ 2 and sufficiently large k, uniformly in r. In particular, $W_r(k)^{1/k}\to\infty$ for each fixed r.

  Formal proof listed

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• 170Sep 24, 2026 Explained

  Sharp logarithmic exponents for off-diagonal Ramsey numbers

  For every fixed integer s ≥ 5, proves $r(s,t)=t^{s-1}/(\log t)^{s-2+o(1)}$ as $t\to\infty$, determining the logarithmic exponent and matching the classical upper bound at that scale. Here $r(s,t)$ is the least number of vertices forcing an s-clique or a t-vertex independent set.

  Formal proof listed

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• 171Sep 23, 2026 Explained

  The hypercube Ramsey conjecture

  Resolves the Burr–Erdős hypercube Ramsey conjecture: the two-color Ramsey number of the n-dimensional cube is $\Theta(2^n)$. Thus every red-blue coloring of a complete graph on a universal constant times the cube's number of vertices contains a monochromatic copy of the cube.

  Claimed, not checked

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• 172Sep 23, 2026 Explained

  Classification of finite Euclidean Ramsey configurations

  Classifies finite point configurations that occur monochromatically, at their original scale, in every finite coloring of sufficiently high-dimensional Euclidean space. The characterization is an algebraic condition over the coordinate field. It also disproves the Leader–Russell–Walters conjecture that every such configuration is a subset of a finite transitive set.

  Formal proof listed

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• 189Sep 25, 2026 Explained

  Cycle–clique Ramsey numbers

  Proves the Erdős–Faudree–Rousseau–Schelp conjecture: $R(C_m,K_n)=(m-1)(n-1)+1$ for every $m\ge n\ge3$, except $R(C_3,K_3)=6$. This is the exact threshold forcing a red m-cycle or a blue n-clique in every red–blue coloring of a complete graph.

  Formal proof listed

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