ar X iv : 0 70 8 . 22 95 v 1 [ m at h . G R ] 1 6 A ug 2 00 7 Product - free subsets of groups , then and now

Let G be a group. A subset S of G is product-free if there do not exist a, b, c ∈ S (not necessarily distinct) 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.) The purpose of this paper is threefold. I first review the history of this problem, up to and including my involvement via Joe Gallian’s REU (Research Experience for Undergraduates) at the University of Minnesota, Duluth, in 1994; since I did this once already in [11], I will be briefer here. I then describe some very recent progress made by Gowers [7]. Finally, I speculate on the gap between the lower and upper bounds, and revisit my 1994 argument to show that this gap cannot be closed using Gowers’s argument as given. Note the usual convention that multiplication and inversion are permitted to act on subsets of G, i.e., for A,B ⊆ G, AB = {ab : a ∈ A, b ∈ B}, A = {a : a ∈ A}.