On families with bounded matching number

Let $e (n, s)$ denote the maximum size of a family of subsets of an $n$ -set without $s$ pairwise disjoint members. The problem of determining $e (n, s)$ was raised by Erdős more than half a century ago. Nevertheless except for the residue classes $n \equiv 0, - 1, - 2 (\mod s)$ the exact value of $e (n, s)$ is largely unknown. In the present paper $e (2 s + r, s)$ is determined for most values of $1 \leq r \leq s$ (see Theorems 1.8 and 1.9). The extremal families are so-called threshold families (see Definition 1.4).

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Keywords

finite sets, matchings, threshold families

Mathematical Subject Classification

Primary: 05D05

Secondary: 05C35

Milestones

Received: 8 March 2023

Accepted: 23 August 2023

Published: 23 September 2023

Authors

Peter Frankl
Rényi Institute Hungarian Academy of Sciences Budapest Hungary

Jian Wang
Department of Mathematics Taiyuan University of Technology Taiyuan China

Vol. 12, No. 3, 2023

Peter Frankl and Jian Wang

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification

Milestones

Authors