Van der Waerden numbers

The quantity is the coefficient $c$ in $W_r(k)>k^{ck\lfloor\log_2r\rfloor}$, for sufficiently large $k$, uniformly in $r$. A higher coefficient is better. The claim assumes the upstream geometric and counting arguments, shorter dilation, local-lemma balance and separate balancing of light positions. The sharper region allows every fixed $c<1/24$; $1/24$ is not an attained value.

OpenAI baseline: $1/100000$, released Oct 6, 2:58 PM PT.

These results are Claimed: their authors published them, and we have not checked them.