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. and Erdos conjectured in 1990 that the lumber 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 In is 2(O(n/m))([(n/2)), for every 1 to [17/21. For m >= root T., this result is sharp up to the constant implicit in the OH. Our proof uses a general bound on the number of independent sets of size to in 3-uniform hypergraphs, proved recently by the authors, and new bounds on the number of integer partitions with small sumset.

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