Beyond sum-free sets in the natural numbers

For an interval [1,N]⊆N, sets S⊆[1,N] with the property that |{(x,y)∈S²:x+y∈S}|=0, known as sum-free sets, have attracted considerable attention. In this paper, we generalize this notion by considering r(S)=|{(x,y)∈S²:x+y∈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.