# 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 $\{1, \dots , n\}$ is much smaller than the number of sum-free sets. In the same paper they gave a lower bound of $2^{\lfloor n/4 \rfloor }$ for the number of maximal sum-free sets. Here, we prove the following: For each $1\leq i \leq 4$, there is a constant $C_i$ such that, given any $n\equiv i \mod 4$, $\{1, \dots , n\}$ contains $(C_i+o(1)) 2^{n/4}$ 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