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

József Balogh, Hong Liu, Maryam Sharifzadeh, Andrew Treglown

Research output: Contribution to journalArticle

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, 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 languageEnglish (US)
Pages (from-to)1885-1911
Number of pages27
JournalJournal of the European Mathematical Society
Volume20
Issue number8
DOIs
StatePublished - 2018

Keywords

  • Container method
  • Independent sets
  • Sum-free sets

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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