The maximum size of intersecting and union families of sets

We consider the maximal size of families of k -element subsets of an n element set [ n ] that satisfy the properties that every r subsets of the family have non-empty intersection, and no ℓ subsets contain [ n ] in their union. We show that for large enough n , the largest such family is the trivial...

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Bibliographic Details
Published inEuropean journal of combinatorics Vol. 33; no. 2; pp. 128 - 138
Main Authors Siggers, Mark, Tokushige, Norihide
Format Journal Article
LanguageEnglish
Published Elsevier Ltd 01.02.2012
Subjects
Combinatorial analysis
Intersections
Unions
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Summary:We consider the maximal size of families of k -element subsets of an n element set [ n ] that satisfy the properties that every r subsets of the family have non-empty intersection, and no ℓ subsets contain [ n ] in their union. We show that for large enough n , the largest such family is the trivial one of all ( n − 2 k − 1 ) subsets that contain a given element and do not contain another given element. Moreover we show that unless such a family is such that all subsets contain a given element, or all subsets miss a given element, then it has size at most . 9 ( n − 2 k − 1 ) . We also obtain versions of these statements for weighted non-uniform families.
Bibliography:ObjectType-Article-2
SourceType-Scholarly Journals-1
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content type line 23
ISSN:0195-6698
1095-9971
DOI:10.1016/j.ejc.2011.08.004