Sum-free sets in abelian groups

TL;DR

This paper characterizes the maximum size and number of sum-free subsets in any abelian group, providing precise asymptotics and bounds that depend on the group's structure.

Contribution

It determines the exact maximum size of sum-free subsets in any abelian group and estimates their total count, including asymptotic formulas for specific group classes.

Findings

Maximum sum-free subset size is c(G)|G| with c(G) in [2/7, 1/2]

Number of sum-free subsets grows as 2^{c(G)|G| + o(|G|)}

Asymptotic formulas are provided for groups with order divisible by certain primes

Abstract

Let A be a subset of an abelian group G. We say that A is sum-free if there do not exist x,y and z in A satisfying x + y = z. We determine, for any G, the cardinality of the largest sum-free subset of G. This equals c(G)|G| where c(G) is a constant depending on G and lying in the interval [2/7,1/2]. We also estimate the number of sum-free subsets of G. It turns out that log_2 of this number is c(G)|G| + o(|G|), which is tight up to the o-term. For certain abelian groups, those whose order is divisible by a small prime of the form 3k + 2, we can obtain an asymptotic for the number of sum-free sets.