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. (Lean)
Manuscript
Quantitative Superexponential Bounds for van der Waerden Numbers
September 23, 2026 · source · Lean formalization available · cite as \cite{OAI:Quantitative-Superexponential-Bounds-for-van-der-Waerden-Numbers-September-23-2026}
We prove that there are absolute constants c > 0 and K0 such that W_r(k)\gt k^{ck\lfloor\log_2 r\rfloor} for every k\ge K_0 and r ≥ 2. Consequently W_r(k)^{1/k}\to\infty for each fixed r ≥ 2, giving a quantitative positive resolution of Erdős's superexponential-growth question, including the two-color case.
Result 160 · Combinatorics. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Agora members.
Source: https://github.com/openai/math/blob/main/CONTENTS.md