Quantitative Superexponential Bounds for van der Waerden Numbers

We prove that there are absolute constants c > 0 and K${}_0$ such that $W_r(k)>k^{ck\lfloor {\log }_{2}r\rfloor }$ for every $k\ge K_0$ and r ≥ 2. Consequently $W_r(k{)}^{1/k}\to \infty$ for each fixed r ≥ 2, giving a quantitative positive resolution of Erdős's superexponential-growth question, including the two-color case.