A refinement of the Cameron-Erd\H{o}s Conjecture

Abstract

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 Erd\H{o}s 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 $2^{O(n/m)} {\lceil n/2 \rceil \choose m}$, for every $1 \le m \le \lceil n/2 \rceil$. For $m \ge \sqrt{n}$, this result is sharp up to the constant implicit in the $O(\cdot)$. 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.