Sharp bound on the number of maximal sum-free subsets of integers
József BaloghUniversity of Illinois at Urbana-Champaign, USA
Hong LiuUniversity of Warwick, Coventry, UK
Maryam SharifzadehUniversity of Warwick, Coventry, UK
Andrew TreglownUniversity of Birmingham, UK
Cameron and Erdős  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  and a recent bound on the number of subsets of integers with small sumset by Green and Morris . 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. 1885–1911DOI 10.4171/JEMS/802