An Application of Lovász's Local Lemma - A New Lower Bound for the van der Waerden Number

The van der Waerden number W(n) is the smallest integer so that if we divide the integers {1,2, …, W(n)} into two classes, then at least one of them contains an arithmetic progression of length n. We prove in this paper that W(n) ≥ 2n/nϵ for all sufficiently large n. © 1990 Wiley Periodicals, Inc.