A refinement of the Cameron-Erds 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 Erds conjectured in 1990 that the number of such sets is O(2^n/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, for every 1 ≤ m ≤ ⌈n/2⌉. For, 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.