Superexponential van der Waerden numbers at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Superexponential van der Waerden numbers

at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Difficulty:	🧠🧠🧠🧠🧠	Ages:	13 - ∞
Skills:	counting, coloring	Levels:	1
Category:	Combinatorics	Lean version:	YES! ✔
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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.

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LEVEL 1 Quantitative Superexponential Bounds for van der Waerden Numbers

released 2026-09-23 | 5 theorems · 15 lemmas · 23 proofs · 10,781 words | PLAY LEVEL 1 » (pdf)

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.

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