Combinatorics · Oct 6, 2026

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.

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