On maximal sum-free sets in abelian groups

Nathanael Hassler, Andrew Treglown

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Abstract

Balogh, Liu, Sharifzadeh and Treglown [Journal of the European Mathematical Society, 2018] recently gave a sharp count on the number of maximal sum-free subsets of {1, . . . , n}, thereby answering a question of Cameron and Erd ̋os. In contrast, not as much is known about the analogous problem for finite abelian groups. In this paper we give the first sharp results in this direction, determining asymptotically the number of maximal sum-free sets in both the binary and ternary spaces Zk2 and Zk3. We also make progress on a conjecture of Balogh, Liu, Sharifzadeh and Treglown concerning a general lower bound on the number of maximal sum-free sets in abelian groups of a fixed order. Indeed, we verify the conjecture for all finite abelian groups with a cyclic component of size at least 3084. Other related results and open problems are also presented.
Original languageEnglish
Article numberP2.32
Pages (from-to)1-24
Number of pages24
JournalThe Electronic Journal of Combinatorics
Volume29
Issue number2
DOIs
Publication statusPublished - 20 May 2022

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics
  • Applied Mathematics

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