On the size of the product of overlapping families

For integers $n > k > 1$ let $(\binom{[n]}{k})$ denote the collection of all $k$ -subsets of the standard $n$ -element set. For $s \geq 2$ and families $ℱ_{i} \subset (\binom{[n]}{k_{i}})$ , $1 \leq i \leq s$ , $(F_{1}, \dots, F_{s})$ is called a rainbow matching if $F_{i} \in ℱ_{i}$ and the $F_{i}$ are pairwise disjoint. Theorem 1.5 provides the best possible upper bounds for the product of the sizes of $ℱ_{i}$ if $n$ is sufficiently large and they span no rainbow matching. For the case of graphs $(k = 2)$ some sharper bounds are established.

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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

Vol. 11, No. 3, 2022

Péter Frankl

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification

Milestones

Authors