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.