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

Cameron and Erd\H{o}s 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 $2^{\lfloor n\mathrm{/}4\rfloor }$ for the number of maximal sum-free sets. Here, we prove the following: For each $1\leq i \leq 4$,thereisaconstant$C_i$suchthat,givenany$n\equiv i \mod 4$,${1, \dots , n}$contains$(C_i+o(1)) 2^{n/4}$ maximal sum-free sets. Our proof makes use of container and removal lemmas of Green, a structural result of Deshouillers, Freiman, S'os 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.