### Abstract

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 language | English (US) |
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Pages (from-to) | 309-344 |

Number of pages | 36 |

Journal | Israel Journal of Mathematics |

Volume | 199 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2014 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

*Israel Journal of Mathematics*,

*199*(1), 309-344. https://doi.org/10.1007/s11856-013-0067-y