Arithmetic progressions in sets with small sumsets

Abstract

We present an elementary proof that if $A$ is a finite set of numbers, and the sumset $A{+}_{G}A$ is small, $|A{+}_{G}A|\le c|A|$, along a dense graph $G$, then $A$ contains $k$-term arithmetic progressions.

To appear in Combinatorics Probability and Computation