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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
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Official URL: https://doi.org/10.4171/JEMS/802
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 | ||||||||||||||||||
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| 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: |
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| 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 | ||||||||||||||||||
| RIOXX Funder/Project Grant: |
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