On compactifications and product-free sets

TL;DR

This paper uses model theoretic methods to analyze the existence and size of product-free sets in groups, providing new conditions and answers to longstanding questions in group theory.

Contribution

It introduces a model theoretic framework to determine when large product-free sets exist in groups, extending previous results by Gowers and others.

Findings

Negative answer to Babai and Sós question on large product-free sets

Model theoretic condition for the existence of large product-free sets

Model theoretic account of Nikolov and Pyber's result on triple products

Abstract

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.