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

Cameron and Erdos [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 equivalent to i mod 4, {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, S6s 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.