Counting sum-free sets in abelian groups

Noga Alon, József Balogh, Robert Morris, Wojciech Samotij

Research output: Contribution to journalArticlepeer-review

24 Scopus citations


In this paper we study sum-free sets of order m in finite abelian groups. We prove a general theorem about independent sets in 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 n is divisible by a prime q with q ≡ 2 (mod 3), for every m ⩾ (Formula presented.), thus extending and refining a theorem of Green and Ruzsa. In particular, we prove that almost all sumfree 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 a fixed size in an (n, d, λ)-graph.

Original languageEnglish (US)
Pages (from-to)309-344
Number of pages36
JournalIsrael Journal of Mathematics
Issue number1
StatePublished - Jan 1 2014
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • General Mathematics


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