Beyond sum-free sets in the natural numbers

TL;DR

This paper explores the properties and possible sum-related values of subsets within a natural number interval, providing a detailed classification and structural insights into extremal sum-free sets.

Contribution

It introduces a comprehensive analysis of the spectrum of sum-related values for subsets of [1,N], including structural characterizations and existence results.

Findings

Complete description of attainable r-values for subsets of [1,N]

Structural characterizations of extremal and near-extremal sets

Constructive existence results for specific sum-related set configurations

Abstract

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.