Counting sum-free sets in Abelian groups

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 $q\equiv 2(\mathrm{mod}3)$, 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.

27 pages

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