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On families with bounded matching number

Peter Frankl and Jian Wang

Vol. 12 (2023), No. 3, 197–222
Abstract

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 0,1,2(mods) the exact value of e(n,s) is largely unknown. In the present paper e(2s + r,s) is determined for most values of 1 r s (see Theorems 1.8 and 1.9). The extremal families are so-called threshold families (see Definition 1.4).

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