Beyond sum-free sets in the natural numbers (Q405098)

Summary: For an interval \([1,N] \subseteq \mathbb{N}\), sets \(S \subseteq [1,N]\) with the property that \(|\{(x,y) \in S^2:x+y \in S\}|=0\), known as sum-free sets, have attracted considerable attention. In this paper, we generalize this notion by considering \(r(S)=|\{(x,y) \in S^2: x+y \in S\}|\), and analyze its behaviour as \(S\) ranges over the subsets of \([1,N]\). We obtain a comprehensive description of the spectrum of attainable \(r\)-values, constructive existence results and structural characterizations for sets attaining extremal and near-extremal values.