On compactifications and product-free sets

A subset of a group is said to be product-free if it does not contain three elements satisfying the equation $xy=z$. We give a negative answer to a question of Babai and S'os on the existence of large product-free sets 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.