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 2bn/4c for the number of maximal sum-free sets. Here, we prove the following: For each 1 ≤ i ≤ 4, there is a constant Ci such that, given any n ≡ i mod 4, {1, . . . , n} contains (Ci + o(1))2 n/4 maximal sum-free sets. Our proof makes use of container and removal lemmas of Green [10, 11], 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 [12]. We also discuss related results and open problems on the number of maximal sum-free subsets of abelian groups.