Product-Free Subsets of Groups, Then and Now

• Main Author: Kedlaya, Kiran S. (Contributor)
• Other Authors: Massachusetts Institute of Technology. Department of Mathematics (Contributor)
• Format: Article
• Language: English
• Published: American Mathematical Society, 2011-06-21T15:27:21Z.
• Subjects: Article
• Online Access: Get fulltext

Dedicated to Joe Gallian on his 65th birthday and the 30th anniversary of the Duluth REU
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.)