A subexponential upper bound for van der Waerden numbers W(3,k)

Abstract: We show an improved upper estimate for van der Waerden number $W(3,k):$ there is an absolute constant $c>0$ such that if ${1,\dots,N}=X\cup Y$ is a partition such that $X$ does not contain any arithmetic progression of length $3$ and $Y$ does not contain any arithmetic progression of length $k$ then $$N\le \exp(O(k^{1-c}))\,.$$