A refinement of the Cameron-Erdős 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 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 $2^{O(n/m)}\left(\genfrac{}{}{0ex}{}{\lceil n/2\rceil }{m}\right)$, 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.

32 pages

Cited by in corpus (2)

• On the structure of large sum-free sets of integers
• The sharp threshold for maximum-size sum-free subsets in even-order abelian groups