The Erdos-Ko-Rado theorem for perfect matchings

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 $k\ge 3t/2+1$, 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 $t\ge 6$. The method used is this paper is similar to the proof of the complete Erdős-Ko-Rado theorem given by Ahlswede and Khachatrian.

arXiv commentThis paper has been withdrawn by the author due to an error in Theorem 2.1, an important case was not considered in this theorem