The Erdos-Ko-Rado theorem for perfect matchings

TL;DR

This paper extends the Erdős-Ko-Rado theorem to perfect matchings in complete graphs, establishing the maximum size of t-intersecting matchings and characterizing extremal systems for certain parameters.

Contribution

It provides a new combinatorial bound for t-intersecting perfect matchings in complete graphs and characterizes the extremal systems achieving this bound.

Findings

Maximum size of t-intersecting 2k-matchings established

Characterization of extremal systems containing t disjoint edges

Bound on k is sharp for t ≥ 6

Abstract

A $2k$-matching is a perfect matching of the complete graph on $2k$ vertices. Two $2k$-matchings are defined to be $t$-intersecting if they have at least $t$ edges in common. The main result in this paper is that if $k \geq 3t/2+1$, then the largest system of $t$-intersecting $2k$-matchings has size $(2(k-t)-1)!! = \prod_{i=0}^{k-t-1}(2k-2t-2i-1)$ and the only systems that meet this bound consist of all $2k$-matchings that contain a set of $t$ disjoint edges. Further, this bound on $k$ is sharp for $t\geq 6$. The method used is this paper is similar to the proof of the complete Erd\H{o}s-Ko-Rado theorem given by Ahlswede and Khachatrian.