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On the size of the product of overlapping families

Péter Frankl

Vol. 11 (2022), No. 3, 237–246
Abstract

For integers n > k > 1 let [n] k denote the collection of all k-subsets of the standard n-element set. For s 2 and families i [n] ki , 1 i s, (F1,,Fs) is called a rainbow matching if Fi i and the Fi are pairwise disjoint. Theorem 1.5 provides the best possible upper bounds for the product of the sizes of i if n is sufficiently large and they span no rainbow matching. For the case of graphs (k = 2) some sharper bounds are established.

Keywords
finite sets, matchings, Erdős–Ko–Rado
Mathematical Subject Classification
Primary: 05D05
Milestones
Received: 1 February 2022
Revised: 9 June 2022
Accepted: 23 June 2022
Published: 15 October 2022
Authors
Péter Frankl
Renyi Institute
Hungarian Academy of Sciences
Budapest
Hungary