Product-Free Subsets of Groups, Then and Now

Abstract

Let G be a group. A subset S of G is product-free if there do not exist a, b, c ∈ S (not necessarily distinct1) such that ab = c. One can ask about the existence of large product-free subsets for various groups, such as the groups of integers (see next section), or compact topological groups (as suggested in [11]). For the rest of this paper, however, I will require G to be a finite group of order n > 1. Let α(G) denote the size of the largest product-free subset of G; put β(G) = α(G)/n, so that β(G) is the density of the largest product-free subset. What can one say about α(G) or β(G) as a function of G, or as a function of n? (Some of our answers will include an unspecified positive constant; I will always call this constant c.)