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Sharp bound on the number of maximal sum-free subsets of integers

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Balogh, József, Liu, Hong, Sharifzadeh, Maryam and Treglown, Andrew (2018) Sharp bound on the number of maximal sum-free subsets of integers. Journal of the European Mathematical Society, 20 (8). pp. 1885-1911. doi:10.4171/JEMS/802 ISSN 1435-9855.

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Official URL: https://doi.org/10.4171/JEMS/802

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Abstract

Cameron and Erdős [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 2 ⌊ n/ 4 ⌋ for the number of maximal sum-free sets. Here, we prove the follo wing: 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 rem oval 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 Gr een and Morris [13]. We also discuss related results and open problems on the number of max imal sum-free subsets of abelian groups.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH): Numbers, Natural
Journal or Publication Title: Journal of the European Mathematical Society
Publisher: European Mathematical Society Publishing House
ISSN: 1435-9855
Official Date: 4 June 2018
Dates:
DateEvent
4 June 2018Published
6 May 2018Accepted
Volume: 20
Number: 8
Page Range: pp. 1885-1911
DOI: 10.4171/JEMS/802
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access
Date of first compliant deposit: 6 June 2018
Date of first compliant Open Access: 6 June 2018
RIOXX Funder/Project Grant:
Project/Grant IDRIOXX Funder NameFunder ID
DMS-1500121National Science Foundationhttp://dx.doi.org/10.13039/100000001
15006Arnold and Mabel Beckman Foundationhttp://dx.doi.org/10.13039/100000997
ECF-2016-5 23Leverhulme Trusthttp://dx.doi.org/10.13039/501100000275
752426H2020 European Research Councilhttp://dx.doi.org/10.13039/100010663
EP/M016641/1[EPSRC] Engineering and Physical Sciences Research Councilhttp://dx.doi.org/10.13039/501100000266
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