{"title":"Maximal k-Sum-Free Collections in an Abelian Group","authors":"Vahe Sargsyan","doi":"10.1134/s1054661824010188","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>Let <span>\\(G\\)</span> be an Abelian group of order <i>n</i>, let <span>\\(k \\geqslant 2\\)</span> be an integer, and <span>\\({{A}_{1}}, \\ldots ,{{A}_{k}}\\)</span> be nonempty subsets of <span>\\(G\\)</span>. The collection <span>\\(\\left( {{{A}_{1}}, \\ldots ,{{A}_{k}}} \\right)\\)</span> is called <span>\\(k\\)</span>-sum-free (abbreviated <span>\\(k\\)</span>-<i>SFC</i>) if the equation <span>\\({{x}_{1}} + \\ldots + {{x}_{k}} = 0\\)</span> has no solutions in the collection <span>\\(\\left( {{{A}_{1}}, \\ldots ,{{A}_{k}}} \\right),\\)</span> where <span>\\({{x}_{1}} \\in {{A}_{1}}\\)</span>, …, <span>\\({{x}_{k}} \\in {{A}_{k}}\\)</span>. The family of <span>\\(k\\)</span>-<i>SFC</i> in <span>\\(G\\)</span> will be denoted by <span>\\(SF{{C}_{k}}\\left( G \\right)\\)</span>. The collection <span>\\(\\left( {{{A}_{1}}, \\ldots ,{{A}_{k}}} \\right) \\in SF{{C}_{k}}\\left( G \\right)\\)</span> is called maximal by capacity if it is maximal by the sum of <span>\\(\\left| {{{A}_{1}}} \\right| + \\ldots + \\left| {{{A}_{k}}} \\right|\\)</span>, and maximal by inclusion if for any <span>\\(i \\in \\left\\{ {1,...,k} \\right\\}\\)</span> and <span>\\(x \\in G{\\kern 1pt} {{\\backslash }}{\\kern 1pt} {{A}_{i}},\\)</span> the collection <span>\\(\\left( {{{A}_{1}},...,{{A}_{{i - 1}}},{{A}_{i}} \\cup \\left\\{ x \\right\\},{{A}_{{i + 1}}},...,{{A}_{k}}} \\right)\\)</span> <span>\\( \\notin \\)</span> <span>\\(SF{{C}_{k}}\\left( G \\right).\\)</span> Suppose <span>\\({{\\varrho }_{k}}\\left( G \\right) = \\left| {{{A}_{1}}} \\right| + \\ldots + \\left| {{{A}_{k}}} \\right|.\\)</span> In this work, we study the problem of the maximal value of <span>\\({{\\varrho }_{k}}\\left( G \\right)\\)</span>. In particular, the maximal value of <span>\\({{\\varrho }_{k}}\\left( {{{Z}_{d}}} \\right)\\)</span> for the cyclic group <span>\\({{Z}_{d}}\\)</span> is determined. Upper and lower bounds for <span>\\({{\\varrho }_{k}}\\left( G \\right)\\)</span> are obtained for the Abelian group <span>\\(G.\\)</span> The structure of the maximal <i>k</i>-sum-free collection by capacity (by inclusion) is described for an arbitrary cyclic group.</p>","PeriodicalId":35400,"journal":{"name":"PATTERN RECOGNITION AND IMAGE ANALYSIS","volume":"43 1","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2024-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"PATTERN RECOGNITION AND IMAGE ANALYSIS","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1134/s1054661824010188","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 0
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.
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