New lower bounds for van der Waerden numbers

We 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(log k/1og log k)(1/3). Previously it had been speculated, supported by data, that w(3, k) = O(k(2)).