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arXiv · 2008.08503

The Erd\H{o}s-Ko-Rado theorem for $2$-intersecting families of perfect matchings

Shaun Fallat·Karen Meagher·Mahsa N. Shirazi
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Abstract

A perfect matching in the complete graph on $2k$ vertices is a set of edges such that no two edges have a vertex in common and every vertex is covered exactly once. Two perfect matchings are said to be $t$-intersecting if they have at least $t$ edges in common. The main result in this paper is an extension of the famous Erd\H{o}s-Ko-Rado (EKR) theorem \cite{EKR} to 2-intersecting families of perfect matchings for all values of $k$. Specifically, for $k\geq 3$ a set of 2-intersecting perfect matchings in $K_{2k}$ of maximum size has $(2k-5)(2k-7)\cdots (1)$ perfect matchings.

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Shaun Fallat, Karen Meagher, Mahsa N. Shirazi. 2020-08-19. The Erd\H{o}s-Ko-Rado theorem for $2$-intersecting families of perfect matchings. https://arxiv.org/abs/2008.08503

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