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The number of maximal sum-free subsets of integers


Authors: József Balogh, Hong Liu, Maryam Sharifzadeh and Andrew Treglown
Journal: Proc. Amer. Math. Soc. 143 (2015), 4713-4721
MSC (2010): Primary 11B75; Secondary 05C69
DOI: https://doi.org/10.1090/S0002-9939-2015-12615-9
Published electronically: April 2, 2015
MathSciNet review: 3391030
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Abstract: Cameron and Erdős raised the question of how many maximal sum-free sets there are in $ \{1, \dots , n\}$, giving a lower bound of $ 2^{\lfloor n/4 \rfloor }$. In this paper we prove that there are in fact at most $ 2^{(1/4+o(1))n}$ maximal sum-free sets in $ \{1, \dots , n\}$. Our proof makes use of container and removal lemmas of Green as well as a result of Deshouillers, Freiman, Sós and Temkin on the structure of sum-free sets.


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József Balogh
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801 — and — Bolyai Institute, University of Szeged, Szeged, Hungary
Email: jobal@math.uiuc.edu

Hong Liu
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
Email: hliu36@illinois.edu

Maryam Sharifzadeh
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
Email: sharifz2@illinois.edu

Andrew Treglown
Affiliation: School of Mathematics, University of Birmingham, United Kingdom
Email: a.c.treglown@bham.ac.uk

DOI: https://doi.org/10.1090/S0002-9939-2015-12615-9
Received by editor(s): May 12, 2014
Received by editor(s) in revised form: July 19, 2014, July 22, 2014, and August 25, 2014
Published electronically: April 2, 2015
Additional Notes: The authors’ research was partially supported by Simons Fellowship, NSF CAREER Grant DMS-0745185, Arnold O. Beckman Research Award (UIUC Campus Research Board 13039) and Marie Curie FP7-PEOPLE-2012-IIF 327763.
Communicated by: Patricia L. Hersh
Article copyright: © Copyright 2015 American Mathematical Society