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

Abstract

Cameron and Erdős asked whether the number of \emph{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 $1\le i\le 4$, there is a constant $C_i$ such that, given any $n\equiv i\mathrm{mod}4$, $\{1,\dots ,n\}$ contains $(C_i+o(1))2^{n/4}$ maximal sum-free sets. Our proof makes use of container and removal lemmas of Green, 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.

arXiv comment25 pages, to appear in the Journal of the European Mathematical Society