Independent Arithmetic Progressions in Clique-Free Graphs on the Natural Numbers

We show that if G is a Kr-free graph on N, there is an independent set in G which contains an arbitrarily long arithmetic progression together with its difference. This is a common generalization of theorems of Schur, van der Waerden, and Ramsey. We also discuss various related questions regarding (m, p, c)-sets and parameter words.