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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.

Manuscripts in this family · 1

Sep 23, 2026·paper.pdfLean

We prove that there are absolute constants c > 0 and K

_{0}

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.

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