Estimates for Cameron–Erdős constants

A set $B$ of integers is called sum-free if for any $a,b \in B$ the number $a+b$ does not belong to the set $B$. Let $s(n)$ be the number of sum-free sets in the interval of natural numbers $[1,n]$. As shown by Cameron, Erdős, and Sapozhenko, there exist constants $c_0$ and $c_1$ such that $s(n)\sim (c_0+1)2^{\lceil n/2\rceil}$ for even $n$ and $s(n)\sim (c_1+1)2^{\lceil n/2\rceil}$ for odd $n$ tending to infinity. The constants $c_0$ and $c_1$ are usually referred to as the Cameron–Erdős constants. In this paper, we obtain upper and lower bounds for the Cameron–Erdős constants which give the two first decimal places of their exact values.
This research was supported by the Russian Foundation for Basic Research, grant 04–01–00359.