Erdos Problem #138: do the van der Waerden numbers grow superexponentially, W(k)^(1/k) -> infinity?

Theorem 1.1: there is an absolute $K_0$ with $W_r(k)>k^{ck\lfloor\log_2 r\rfloor}$, $c=10^{-5}$, for all $k\ge K_0$ and $r\ge2$; hence $W_r(k)^{1/k}\to\infty$ for each fixed $r$. Prior work in 2026: Fox-Hunter proved superexponential growth for three colours, and Campos-Fox-Schildkraut proved $W_2(k)\ge(1-o(1))k2^{k-1}$, settling $W_2(k)/2^k\to\infty$ but not the limit asked here. Not shown: the companion question $W(k+1)-W(k)\to\infty$, or any upper bound improvement.

Theorem 1.1: there is an absolute $K_0$ with $W_r(k)>k^{ck\lfloor\log_2 r\rfloor}$, $c=10^{-5}$, for all $k\ge K_0$ and $r\ge2$; hence $W_r(k)^{1/k}\to\infty$ for each fixed $r$. Prior work in 2026: Fox-Hunter proved superexponential growth for three colours, and Campos-Fox-Schildkraut proved $W_2(k)\ge(1-o(1))k2^{k-1}$, settling $W_2(k)/2^k\to\infty$ but not the limit asked here. Not shown: the companion question $W(k+1)-W(k)\to\infty$, or any upper bound improvement.