{"title":"多部分交叉族","authors":"Yuanxiao Xi, Xiangliang Kong, Gennian Ge","doi":"10.1007/s10801-024-01301-6","DOIUrl":null,"url":null,"abstract":"<p>Let <span>\\({\\mathcal {A}}\\subseteq {[n]\\atopwithdelims ()a}\\)</span> and <span>\\({\\mathcal {B}}\\subseteq {[n]\\atopwithdelims ()b}\\)</span> be two families of subsets of [<i>n</i>], we say <span>\\({\\mathcal {A}}\\)</span> and <span>\\({\\mathcal {B}}\\)</span> are cross-intersecting if <span>\\(A\\cap B\\ne \\emptyset \\)</span> for all <span>\\(A\\in {\\mathcal {A}}\\)</span>, <span>\\(B\\in {\\mathcal {B}}\\)</span>. In this paper, we study cross-intersecting families in the multi-part setting. By characterizing the independent sets of vertex-transitive graphs and their direct products, we determine the sizes and structures of maximum-sized multi-part cross-intersecting families. This generalizes the results of Hilton’s (J Lond Math Soc 15(2):369–376, 1977) and Frankl–Tohushige’s (J Comb Theory Ser A 61(1):87–97, 1992) on cross-intersecting families in the single-part setting.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Multi-part cross-intersecting families\",\"authors\":\"Yuanxiao Xi, Xiangliang Kong, Gennian Ge\",\"doi\":\"10.1007/s10801-024-01301-6\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <span>\\\\({\\\\mathcal {A}}\\\\subseteq {[n]\\\\atopwithdelims ()a}\\\\)</span> and <span>\\\\({\\\\mathcal {B}}\\\\subseteq {[n]\\\\atopwithdelims ()b}\\\\)</span> be two families of subsets of [<i>n</i>], we say <span>\\\\({\\\\mathcal {A}}\\\\)</span> and <span>\\\\({\\\\mathcal {B}}\\\\)</span> are cross-intersecting if <span>\\\\(A\\\\cap B\\\\ne \\\\emptyset \\\\)</span> for all <span>\\\\(A\\\\in {\\\\mathcal {A}}\\\\)</span>, <span>\\\\(B\\\\in {\\\\mathcal {B}}\\\\)</span>. In this paper, we study cross-intersecting families in the multi-part setting. By characterizing the independent sets of vertex-transitive graphs and their direct products, we determine the sizes and structures of maximum-sized multi-part cross-intersecting families. This generalizes the results of Hilton’s (J Lond Math Soc 15(2):369–376, 1977) and Frankl–Tohushige’s (J Comb Theory Ser A 61(1):87–97, 1992) on cross-intersecting families in the single-part setting.</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-02-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s10801-024-01301-6\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10801-024-01301-6","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
让({\mathcal {A}}subseteq {[n]\atopwithdelims()a}\)和({\mathcal {B}}subseteq {[n]\atopwithdelims()b}\)是[n]的两个子集族、如果对于所有的\(A in {\mathcal {A}}\),\(B in {\mathcal {B}}\),\(Acap Bne \emptyset \)都是交叉的,我们就说\({\mathcal {A}}\)和\({\mathcal {B}}\)是交叉的。在本文中,我们将研究多部分环境下的交叉相交族。通过描述顶点变换图的独立集及其直接乘积,我们确定了最大尺寸的多部分交叉族的大小和结构。这概括了希尔顿(J Lond Math Soc 15(2):369-376, 1977)和弗兰克尔-托胡希(Frankl-Tohushige)(J Comb Theory Ser A 61(1):87-97, 1992)关于单部分交叉族的结果。
Let \({\mathcal {A}}\subseteq {[n]\atopwithdelims ()a}\) and \({\mathcal {B}}\subseteq {[n]\atopwithdelims ()b}\) be two families of subsets of [n], we say \({\mathcal {A}}\) and \({\mathcal {B}}\) are cross-intersecting if \(A\cap B\ne \emptyset \) for all \(A\in {\mathcal {A}}\), \(B\in {\mathcal {B}}\). In this paper, we study cross-intersecting families in the multi-part setting. By characterizing the independent sets of vertex-transitive graphs and their direct products, we determine the sizes and structures of maximum-sized multi-part cross-intersecting families. This generalizes the results of Hilton’s (J Lond Math Soc 15(2):369–376, 1977) and Frankl–Tohushige’s (J Comb Theory Ser A 61(1):87–97, 1992) on cross-intersecting families in the single-part setting.