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

  • József Balogh

    University of Illinois at Urbana-Champaign, USA
  • Hong Liu

    University of Warwick, Coventry, UK
  • Maryam Sharifzadeh

    University of Warwick, Coventry, UK
  • Andrew Treglown

    University of Birmingham, UK

Abstract

Cameron and Erdős [6] asked whether the number of maximal sum-free sets in is much smaller than the number of sum-free sets. In the same paper they gave a lower bound of for the number of maximal sum-free sets. Here, we prove the following: For each , there is a constant such that, given any , contains 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.

Cite this article

József Balogh, Hong Liu, Maryam Sharifzadeh, Andrew Treglown, Sharp bound on the number of maximal sum-free subsets of integers. J. Eur. Math. Soc. 20 (2018), no. 8, pp. 1885–1911

DOI 10.4171/JEMS/802