### Abstract

In this paper, we study sum-free subsets of the set 1, ..., n}, that is, subsets of the first n positive integers which contain no solution to the equation x+y=z. Cameron and Erds conjectured in 1990 that the number of such sets is O(2^{n/2}). This conjecture was confirmed by Green and, independently, by Sapozhenko. Here, we prove a refined version of their theorem, by showing that the number of sum-free subsets of [n] of size m is, for every 1 ≤ m ≤ ⌈n/2⌉. For, this result is sharp up to the constant implicit in the O(·). Our proof uses a general bound on the number of independent sets of size m in 3-uniform hypergraphs, proved recently by the authors, and new bounds on the number of integer partitions with small sumset.

Original language | English (US) |
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Pages (from-to) | 44-72 |

Number of pages | 29 |

Journal | Proceedings of the London Mathematical Society |

Volume | 108 |

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

*Proceedings of the London Mathematical Society*,

*108*(1), 44-72. https://doi.org/10.1112/plms/pdt033