Problem 138

Erdős Problem #138: declared status 'open'. Formalized: yes. Let the van der Waerden number $W(k)$ be such that whenever $N\geq W(k)$ and $\{1,\ldots,N\}$ is $2$-coloured there must exist a monochromatic $k$-term arithmetic progression. Improve the bounds for $W(k)$ - for example, prove that $W(k)^{1/k}\to \infty$. Current best: Gowers [Go01] has proved$$W(k) \leq 2^{2^{2^{2^{2^{k+9}}}}}.$$The best general lower bound is $W(k)\gg 2^k$, due to Kozik and Shabanov [KoSh16]. Prize: $500. OEIS: A005346. Tags: additive combinatorics.