Erdős Problem #138

The problem

Let the van der Waerden number $W(k)$ be such that whenever $N\geq W(k)$ and $\{1,\ldots,N\}$ is $2$ -coloured there must exist a monochromatic $k$ -term arithmetic progression. Improve the bounds for $W(k)$ - for example, prove that $W(k)^{1/k}\to \infty$ .

Statement quoted from erdosproblems.com/138, maintained by Thomas Bloom. Original source: [Er57][Er61][Er73][Er74b][Er75b][Er77c][ErGr79][Er80,p.90][ErGr80][Er81][Er97c]. Current status there: open.

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Date	Kind	AI systems	Humans	Outcome
10 Apr, 2026	AI on its own	DeepMind prover agent	—	🟡 Partial result (Lean)
10 Apr, 2026	AI working with humans	GPT-5.4 Thinking	Nat Sothanaphan	🟡 Partial result of variant problem

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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