@article{572aa21c8e264ec49dacaa6fc0290758,

title = "Sharp bound on the number of maximal sum-free subsets of integers",

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 Ci such that, given any n ≡ i mod 4, {1, . . ., n} contains (Ci + o(1))2n/4 maximal sum-free sets. Our proof makes use of container and removal lemmas of Green [11, 12], a structural result of Deshouillers, Freiman, S{\'o}s 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.",

keywords = "Container method, Independent sets, Sum-free sets",

author = "J{\'o}zsef Balogh and Hong Liu and Maryam Sharifzadeh and Andrew Treglown",

note = "Funding Information: The authors are grateful to the BRIDGE strategic alliance between the University of Birmingham and the University of Illinois at Urbana-Champaign. This research was conducted as part of the {\textquoteleft}Building Bridges in Mathematics{\textquoteright} BRIDGE Seed Fund project. The work was done while Hong Liu and Maryam Sharifzadeh were postgraduate students at the University of Illinois at Urbana-Champaign. The authors are also grateful to the referees for their careful reviews. Research of J. Balogh is partially supported by NSF Grant DMS-1500121 and an Arnold O. Beckman Research Award (UIUC Campus Research Board 15006). Research of H. Liu is supported by the Leverhulme Trust Early Career Fellowship ECF-2016-523. Research of M. Sharifzadeh is supported by the European Unions Horizon 2020 research and innovation programme under the Marie Curie Individual Fellowship No. 752426. Research of A. Treglown is supported by EPSRC grant EP/M016641/1. Funding Information: Acknowledgments. The authors are grateful to the BRIDGE strategic alliance between the University of Birmingham and the University of Illinois at Urbana-Champaign. This research was conducted as part of the {\textquoteleft}Building Bridges in Mathematics{\textquoteright} BRIDGE Seed Fund project. The work was done while Hong Liu and Maryam Sharifzadeh were postgraduate students at the University of Illinois at Urbana-Champaign. The authors are also grateful to the referees for their careful reviews. Research of J. Balogh is partially supported by NSF Grant DMS-1500121 and an Arnold O. Beckman Research Award (UIUC Campus Research Board 15006). Research of H. Liu is supported by the Leverhulme Trust Early Career Fellowship ECF-2016-523. Research of M. Sharifzadeh is supported by the European Unions Horizon 2020 research and innovation programme under the Marie Curie Individual Fellowship No. 752426. Research of A. Treglown is supported by EPSRC grant EP/M016641/1. Publisher Copyright: {\textcopyright} European Mathematical Society 2018",

year = "2018",

doi = "10.4171/JEMS/802",

language = "English (US)",

volume = "20",

pages = "1885--1911",

journal = "Journal of the European Mathematical Society",

issn = "1435-9855",

publisher = "European Mathematical Society Publishing House",

number = "8",

}