  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
$$N\le \exp(O(k^{1-c}))\,.$$
