On the number and structure of sum-free sets in a segment of positive integers

K. G. Omel'yanov ; A. A. Sapozhenko

Diskretnaya Matematika, Tome 15 (2003) no. 4, pp. 141-147

Résumé

A set $A$ of integers is called sum-free if $a+b\notin A$ for any $a,b\in A$. For any real numbers $q\le p$ we denote by $[q,p]$ the set of real numbers $x$ such that $q\le x\le p$. Let $S(t,n)$ stand for the family of all sum-free subsets $A\subseteq[t,n]$, and $s(t,n)=|S(t,n)|$. We prove that \begin{equation*} s(t,n)=O(2^{n/2}) \end{equation*} for $t\ge n^{3/4}\log n$, where $\log t=\log_2t$. This research was supported by the Russian Foundation for Basic Research, grant 01–01–00266.