Improved lower bounds for multicolor van der Waerden numbers

Given integers k,r, we define the multicolor van der Waerden number w(k;r) to be the largest integer N so that {1,...,N} can be partitioned into r color classes which each lack arithmetic progressions of length k. It is a well-studied problem in additive combinatorics to obtain bounds for this problem. In this talk, we present a new lower bound of the form w(k;r) > (r!-o(1))^k (for fixed r and large k). Some may view this as surprising, as all previous constructions for the multicolor van Waerden numbers closely paralleled lower bounds obtained for the multicolor Ramsey numbers. We achieve this with some new ideas, building upon previous work of the speaker, which further allow for a quick resolution to a question of Erdos (https://www.erdosproblems.com/190).