OpenAI research collection · 1 manuscript · Source checked 2026-10-19
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\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}→∞ for each fixed r.
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We prove that there are absolute constants c > 0 and K_(0) such that W_r(k)> k^{ck\lfloor\log_2 r\rfloor} for every k≥ K_0 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.
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