JournalsjemsVol. 20, No. 8pp. 1885–1911

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

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Abstract

Cameron and Erdős [6] asked whether the number of maximal sum-free sets in {1,,n}\{1, \dots , n\} is much smaller than the number of sum-free sets. In the same paper they gave a lower bound of 2n/42^{\lfloor n/4 \rfloor } for the number of maximal sum-free sets. Here, we prove the following: For each 1i41\leq i \leq 4, there is a constant CiC_i such that, given any nimod4n\equiv i \mod 4, {1,,n}\{1, \dots , n\} contains (Ci+o(1))2n/4(C_i+o(1)) 2^{n/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.

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