New lower bounds for van der Waerden numbers

Abstract

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