Remark: Family 160. Superexponential van der Waerden numbers

Resolves Erdős's superexponential-growth question for van der Waerden numbers. If $W_r(k)$ is the least interval length forcing a monochromatic $k$-term progression in every $r$-coloring, then $W_r(k)>k^{ck\lfloor {\log }_{2}r\rfloor }$ for an absolute $c>0$, all $r\ge 2$ and sufficiently large $k$, uniformly in $r$. In particular, $W_r(k{)}^{1/k}\to \infty$ for each fixed $r$.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.