### Abstract

Let A be a subset of an abelian group G with |G| = n. We say that A is sum-free if there do not exist x, y, z ∈ A with x + y = z. We determine, for any G, the maximal density μ(G) of a sum-free subset of G. This was previously known only for certain G. We prove that the number of sum-free subsets of G is 2^{(μ(G)+o(1))n}, which is tight up to the o-term. For certain groups, those with a small prime factor of the form 3k + 2, we are able to give an asymptotic formula for the number of sum-free subsets of G. This extends a result of Lev, Łuczak and Schoen who found such a formula in the case n even.

Original language | English |
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Pages (from-to) | 157-188 |

Number of pages | 32 |

Journal | Israel Journal of Mathematics |

Volume | 147 |

DOIs | |

Publication status | Published - Jan 1 2005 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Green, B., & Ruzsa, I. Z. (2005). Sum-free sets in abelian groups.

*Israel Journal of Mathematics*,*147*, 157-188. https://doi.org/10.1007/BF02785363