### Abstract

Cameron and Erdos [6] asked whether the number of maximal sum-free sets in {1, . . ., n} is much smaller than the number of sum-free sets. In the same paper they gave a lower bound of 2n/4 for the number of maximal sum-free sets. Here, we prove the following: For each 1 ≤ i ≤ 4, there is a constant C_{i} such that, given any n ≡ i mod 4, {1, . . ., n} contains (C_{i} + o(1))2^{n/}^{4} maximal sum-free sets. Our proof makes use of container and removal lemmas of Green [11, 12], a structural result of Deshouillers, Freiman, Sós and Temkin [7] and a recent bound on the number of subsets of integers with small sumset by Green and Morris [13]. We also discuss related results and open problems on the number of maximal sum-free subsets of abelian groups.

Original language | English (US) |
---|---|

Pages (from-to) | 1885-1911 |

Number of pages | 27 |

Journal | Journal of the European Mathematical Society |

Volume | 20 |

Issue number | 8 |

DOIs | |

State | Published - 2018 |

### Keywords

- Container method
- Independent sets
- Sum-free sets

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

## Fingerprint Dive into the research topics of 'Sharp bound on the number of maximal sum-free subsets of integers'. Together they form a unique fingerprint.

## Cite this

*Journal of the European Mathematical Society*,

*20*(8), 1885-1911. https://doi.org/10.4171/JEMS/802