Maximal k-Sum-Free Collections in an Abelian Group

Abstract

Let \(G\) be an Abelian group of order n, let \(k \geqslant 2\) be an integer, and \({{A}_{1}}, \ldots ,{{A}_{k}}\) be nonempty subsets of \(G\). The collection \(\left( {{{A}_{1}}, \ldots ,{{A}_{k}}} \right)\) is called \(k\)-sum-free (abbreviated \(k\)-SFC) if the equation \({{x}_{1}} + \ldots + {{x}_{k}} = 0\) has no solutions in the collection \(\left( {{{A}_{1}}, \ldots ,{{A}_{k}}} \right),\) where \({{x}_{1}} \in {{A}_{1}}\), …, \({{x}_{k}} \in {{A}_{k}}\). The family of \(k\)-SFC in \(G\) will be denoted by \(SF{{C}_{k}}\left( G \right)\). The collection \(\left( {{{A}_{1}}, \ldots ,{{A}_{k}}} \right) \in SF{{C}_{k}}\left( G \right)\) is called maximal by capacity if it is maximal by the sum of \(\left| {{{A}_{1}}} \right| + \ldots + \left| {{{A}_{k}}} \right|\), and maximal by inclusion if for any \(i \in \left\{ {1,...,k} \right\}\) and \(x \in G{\kern 1pt} {{\backslash }}{\kern 1pt} {{A}_{i}},\) the collection \(\left( {{{A}_{1}},...,{{A}_{{i - 1}}},{{A}_{i}} \cup \left\{ x \right\},{{A}_{{i + 1}}},...,{{A}_{k}}} \right)\) \( \notin \) \(SF{{C}_{k}}\left( G \right).\) Suppose \({{\varrho }_{k}}\left( G \right) = \left| {{{A}_{1}}} \right| + \ldots + \left| {{{A}_{k}}} \right|.\) In this work, we study the problem of the maximal value of \({{\varrho }_{k}}\left( G \right)\). In particular, the maximal value of \({{\varrho }_{k}}\left( {{{Z}_{d}}} \right)\) for the cyclic group \({{Z}_{d}}\) is determined. Upper and lower bounds for \({{\varrho }_{k}}\left( G \right)\) are obtained for the Abelian group \(G.\) The structure of the maximal k-sum-free collection by capacity (by inclusion) is described for an arbitrary cyclic group.