On compactifications and product‐free sets

Abstract

A subset of a group is said to be product free if it does not contain three elements satisfying the equation x·y=z. We give a negative answer to a question of Babai and Sós on the existence of large product-free sets in finite groups by model theoretic means. This question was originally answered by Gowers. Furthermore, we give a natural and sufficient model theoretic condition for a group to have a large product-free subset, as well as a model theoretic account of a result of Nikolov and Pyber on triple products.