A Refinement of the Cameron-erdős Conjecture

In this paper we study sum-free subsets of the set {1, . . . , n}, that is, subsets of the first n positive integers which contain no solution to the equation x+ y = z. Cameron and Erdős conjectured in 1990 that the number of such sets is O(2). This conjecture was confirmed by Green and, independently, by Sapozhenko. Here we prove a refined version of their theorem, by showing that the number of sum-free subsets of [n] of size m is 2 (dn/2e m ) , for every 1 6 m 6 dn/2e. For m > √ n, this result is sharp up to the constant implicit in the O(·). Our proof uses a general bound on the number of independent sets of size m in 3-uniform hypergraphs, proved recently by the authors, and new bounds on the number of integer partitions with small sumset.