Counting sum-free sets in Abelian groups

TL;DR

This paper analyzes the structure and enumeration of sum-free sets in finite Abelian groups, extending previous results and providing new structural and counting theorems using hypergraph and spectral methods.

Contribution

It introduces a general hypergraph theorem linking sparse and dense cases, and determines the typical structure and count of sum-free sets in certain Abelian groups, extending prior work.

Findings

Almost all sum-free subsets of size m are contained in a maximum sum-free subset.

Determined the asymptotic number of sum-free sets in specific Abelian groups.

Provided a new spectral bound on independent sets in graphs.

Abstract

In this paper we study sum-free sets of order $m$ in finite Abelian groups. We prove a general theorem on 3-uniform hypergraphs, which allows us to deduce structural results in the sparse setting from stability results in the dense setting. As a consequence, we determine the typical structure and asymptotic number of sum-free sets of order $m$ in Abelian groups $G$ whose order is divisible by a prime $q$ with $q \equiv 2 \pmod 3$, for every $m \ge C(q) \sqrt{n \log n}$, thus extending and refining a theorem of Green and Ruzsa. In particular, we prove that almost all sum-free subsets of size $m$ are contained in a maximum-size sum-free subset of $G$. We also give a completely self-contained proof of this statement for Abelian groups of even order, which uses spectral methods and a new bound on the number of independent sets of size $m$ in an $(n,d,\lambda)$-graph.