  For an interval [1,N] in the natural numbers, investigating subsets S of
[1,N] such that |{(x,y) in S^2:x+y in S}|=0, known as sum-free sets, has
attracted considerable attention. In this paper, we define r(S):=|{(x,y) in
S^2: x+y in S}| and consider its behaviour as S ranges over the subsets of
[1,N]. We obtain a comprehensive description of the spectrum of attainable
r-values for the s-sets of [1,N], constructive existence results and structural
characterizations for sets attaining extremal and near-extremal values.
