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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  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    2   O  (  n   /   m  )(⌈  n   /   2  ⌉   m    ), for every 1 ⩽  m  ⩽ ⌈ n /2⌉. 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.

A refinement of the Cameron–Erdős conjecture