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

Cameron and Erdős [6] asked whether the number of maximal sum-free sets in {1,…,n} is much smaller than the number of sum-free sets. In the same paper they gave a lower bound of 2^⌊n/4⌋ for the number of maximal sum-free sets. Here, we prove the following: For each 1≤i≤4, there is a constant C_i such that, given any n≡imod4, {1,…,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.