Superexponential van der Waerden numbers

What OpenAI claims

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) > k^{ck⌊log₂ r⌋} for an absolute c > 0, all r ≥ 2 and sufficiently large k, uniformly in r. In particular, W_r(k)^1/k → ∞ for each fixed r. (Lean)

Paper

1. Quantitative Superexponential Bounds for van der Waerden Numbers

   23 September 2026 · Preprint, not peer reviewed · PDF

   We prove that there are absolute constants c > 0 and K0 such that W_r(k) > k^{ck⌊log₂ r⌋} for every k ≥ K₀ and r ≥ 2. Consequently W_r(k)^1/k → ∞ for each fixed r ≥ 2, giving a quantitative positive resolution of Erdős's superexponential-growth question, including the two-color case.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.