ChatGPT 5.6 Sol Pro Finds Erdős's Conjectured van der Waerden Lower Bound

In a preprint first uploaded on August 21, 2026, Marcelo Campos, Jacob Fox, and Carl Schildkraut prove that the two-color van der Waerden number satisfies

$$w(k)\geq(1-o(1))k2^{k-1}$$

for every positive integer $k$. This verifies a conjecture of Erdős. Berlekamp had obtained the same asymptotic lower bound in 1968 when $k-1$ is prime; the new result covers general $k$.

The construction combines Berlekamp-type colorings through a product indexed by suitable primes. It improves the previously quoted general lower bound $w(k)\geq c2^k$ by the asymptotic factor of order $k$, while remaining a lower bound rather than a determination of the exact van der Waerden numbers.

The authors state that a query to ChatGPT 5.6 Sol Pro generated both the coloring and a proof. Their published proof draws partly on that output but is their own modification of an exposition they wrote for Berlekamp's prime case; they state that all writing in the preprint is theirs.