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Erdos Problem #138: do the van der Waerden numbers grow superexponentially, W(k)^(1/k) -> infinity?

Theorem 1.1: there is an absolute $K_0$ with $W_r(k)>k^{ck\lfloor\log_2 r\rfloor}$ , $c=10^{-5}$ , for all $k\ge K_0$ and $r\ge2$ ; hence $W_r(k)^{1/k}\to\infty$ for each fixed $r$ . Prior work in 2026: Fox-Hunter proved superexponential growth for three colours, and Campos-Fox-Schildkraut proved $W_2(k)\ge(1-o(1))k2^{k-1}$ , settling $W_2(k)/2^k\to\infty$ but not the limit asked here. Not shown: the companion question $W(k+1)-W(k)\to\infty$ , or any upper bound improvement.

Exact FrontierDelta

Prior state unknown→proved

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Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Erdos Problem #138: do the van der Waerden numbers grow superexponentially, W(k)^(1/k) -> infinity? · Superexponential van der Waerden numbers

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Registry verification: lean checked · announcement · candidate

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VibeMathed record: Erdos Problem #138: do the van der Waerden numbers grow superexponentially, W(k)^(1/k) -> infinity?

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No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 were read against Erdos's question; the theorem implies $W_r(k)^{1/k}\to\infty$ for each fixed $r\ge2$ , including $r=2$ . Lean: formalization.yaml lists OAI.QuantitativeVanDerWaerden.uniform_lower_bound (OAI/Combinatorics/ProgressionColoring/Main.lean). The comparator statement was read: it asserts an absolute $K$ with $k^{k\lfloor\log_2 r\rfloor/100000}<W(r,k)$ for all $k\ge K$ , $r\ge2$ , where $W$ is defined as an infimum without a built-in finiteness proof (the strict inequality forces finiteness). This is the headline bound; the limit is an immediate consequence. Not rebuilt here.

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