Integer Sets Containing No Arithmetic Progressions

Abstract

lfh and k are positive integers there exists N(h, k) such that whenever N ^ N(h, k), and the integers 1,2,...,N are divided into h subsets, at least one must contain an arithmetic progression of length k. This is the famous theorem of van der Waerden [10], dating from 1927. The proof of this uses multiple nested inductions, which result in extremely weak bounds for N{h,k). We shall define Bk to be the collection of all sets si £= N for which sf contains no arithmetic progression of length k. We then set