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arXiv · 2608.20824

A new lower bound for two-color van der Waerden numbers

Marcelo Campos·Jacob Fox·Carl Schildkraut
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Abstract

The van der Waerden number $w(k)$ is the smallest positive integer $N$ such that every two-coloring of $\{1,2,\ldots,N\}$ contains a monochromatic $k$-term arithmetic progression. We prove that $w(k) \geq (1-o(1))k2^{k-1}$ holds for all positive integers $k$. This verifies a conjecture of Erdős. In 1968, Berlekamp proved the same result when $k-1$ is prime. The coloring for general $k$ can be viewed as a product of Berlekamp's colorings for various primes. It was found by ChatGPT 5.6 Sol Pro.

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Marcelo Campos, Jacob Fox, Carl Schildkraut. 2026-08-21. A new lower bound for two-color van der Waerden numbers. https://arxiv.org/abs/2608.20824

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