Beyond sum-free sets in the natural numbers

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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.

Original languageEnglish
Number of pages20
JournalElectronic Journal of Combinatorics
Issue number1
Publication statusPublished - 7 Feb 2014


  • Sum-free sets


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