The Erdős–Ko–Rado theorem for 2-intersecting families of perfect matchings

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 Erdős-Ko-Rado (EKR) theorem [4] to 2-intersecting families of perfect matchings for all values of k. Specifically, for k 3 a set of 2-intersecting perfect matchings in K 2k of maximum size has (2k − 5)(2k − 7) • • • (1) perfect matchings.If equality holds, then F consists of all k-subsets containing a fixed t-subset of {1, 2, . . . , n}.Twenty-three years after the publication of Erdős, Ko and Rado's work, Wilson [20] enhanced their results by giving an algebraic proof of the their result with the exact value of f (k, t) for all k and t. Later in 1997, Ahlswede and Khachatrian [1] found all maximum t-intersecting families of k-subsets for all values of n. In 2011, Ellis, Friedgut, and Pilpel [3] showed that the analog of the EKR theorem holds for tintersecting families of permutations of {1, . . . , n}, when n is sufficiently large relative to t. In 2005, Meagher and Moura [13] proved that a natural version of the EKR Manuscript

Some Erdös-Ko-Rado results for linear and affine groups of degree two

“…A c c e p t e d m a n u s c r i p t It is known that if a group has a regular subgroup, then there is a weighting of the conjugacy classes so that the ratio bound holds with equality [15,Theorem 3.5]. It would be interesting to know of more cases where it can be shown that such a weighting exists.…”

An extension of the Erdős-Ko-Rado theorem to uniform set partitions

“…This result has motivated consideration of "intersecting" families of many A c c e p t e d m a n u s c r i p t other combinatorial objects using diverse proof techniques and has developed into an active and broad area of research. There are many recent results giving analogs of the EKR theorem; see, for example, [9,13,16,20,26] or [12] and the references within. In this work, we prove an extension of the EKR theorem to systems of uniform set partitions.…”