Three-color van der Waerden numbers grow super-exponentially

For $k$ sufficiently large, we show that there is a three-coloring of the first $2^{k({\log }^{\ast }k)\mathrm{/}4}$ positive integers without any monochromatic $k$-term arithmetic progressions. Thus, the three-color van der Waerden number $w(k;3)$ grows faster than any exponential in $k$. We further prove a new lower bound on multicolor van der Waerden numbers which resolves a problem of Erdős and Graham on canonical van der Waerden numbers.