Improved lower bounds for van der Waerden numbers

Recently, Ben Green proved that the two-color van der Waerden number $w(3,k)$ is bounded from below by $k^{b_0(k)}$ where $b_0(k) = c_0\left(\frac{\log k }{\log \log k}\right)^{1/3}$. We prove a new lower bound of $k^{b(k)}$ with $b(k) = \frac{c\log k}{\log \log k}$. This is done by modifying Green's argument, replacing a complicated result about random quadratic forms with an elementary probabilistic result.