New lower bounds for van der Waerden numbers

Abstract We show that there is a red-blue colouring of $[N]$ with no blue 3-term arithmetic progression and red length $e^{C(\log N)^{3/4}(\log \log N)^{1/4}}$ . Consequently, the two-colour van der Waerden number $w(3,k)$ bounded below by $k^{b(k)}$ , where $b(k) = c \big ( \frac {\log k}{\log k} )^{1/3}$ Previously it had been speculated, supported data, $w(3,k) O(k^2)$