Product-free subsets of groups, then and now

TL;DR

This paper reviews progress on the problem of determining the maximum size of product-free subsets in finite groups, highlighting bounds and their near-optimality, with a focus on recent theoretical advances.

Contribution

The paper provides a comprehensive review of bounds on product-free subsets, including a refined lower bound construction that matches Gowers' upper bound, demonstrating near-optimality.

Findings

Gowers' upper bound is essentially optimal for the generalized problem.

Refined lower bound construction matches Gowers' upper bound.

Progress in understanding the size of product-free subsets in finite groups.

Abstract

A subset of a group is product-free if it does not contain elements a, b, c such that ab = c. We review progress on the problem of determining the size of the largest product-free subset of an arbitrary finite group, including a lower bound due to the author, and a recent upper bound due to Gowers. The bound of Gowers is more general; it allows three different sets A, B, C such that one cannot solve ab = c with a in A, b in B, c in C. We exhibit a refinement of the lower bound construction which shows that for this broader question, the bound of Gowers is essentially optimal.