On maximal sum-free sets in abelian groups

TL;DR

This paper determines the asymptotic number of maximal sum-free sets in binary and ternary abelian groups, extending known results from integers and addressing conjectures about their counts in general finite abelian groups.

Contribution

It provides the first sharp asymptotic results for maximal sum-free sets in finite abelian groups beyond integers, and verifies a conjecture for groups with large cyclic components.

Findings

Asymptotic counts for maximal sum-free sets in ^k and ^k.

Verification of a conjecture for groups with cyclic component 08084 or larger.

Progress on lower bounds for the number of maximal sum-free sets in abelian groups.

Abstract

Balogh, Liu, Sharifzadeh and Treglown [Journal of the European Mathematical Society, 2018] recently gave a sharp count on the number of maximal sum-free subsets of $\{1, \dots, n\}$, thereby answering a question of Cameron and Erd\H{o}s. In contrast, not as much is know about the analogous problem for finite abelian groups. In this paper we give the first sharp results in this direction, determining asymptotically the number of maximal sum-free sets in both the binary and ternary spaces $\mathbb Z^k_2$ and $\mathbb Z^k_3$. We also make progress on a conjecture of Balogh, Liu, Sharifzadeh and Treglown concerning a general lower bound on the number of maximal sum-free sets in abelian groups of a fixed order. Indeed, we verify the conjecture for all finite abelian groups with a cyclic component of size at least 3084. Other related results and open problems are also presented.