Abstract
The Erdős–Ko–Rado theorem for 2-intersecting families of perfect matchings
Highlights
Introduction and PreliminariesIn this paper we present two different approaches to establish a version of the Erdős– Ko–Rado theorem for 2-intersecting families of perfect matchings
Our goal is to show that the set of perfect matchings with two fixed edges is a maximum coclique in M2(2k)
For 10 k 14, similar to the proof of Theorem 4.10, by utilizing the complete character tables of the perfect matching association scheme [14, 16] we can find all the eigenvalues of the matrix M, and we see that the ratio bound holds with equality
Summary
In this paper we present two different approaches to establish a version of the Erdős– Ko–Rado theorem for 2-intersecting families of perfect matchings. In 1961, they proved if F is a t-intersecting family of k-subsets of {1, 2, . N}, there is a tight upper bound on the size of F with n sufficiently large [4]. Theorem 1.1 (EKR, [4]). If F is a t-intersecting family of k-subsets of {1, 2, . 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, . Erdős–Ko–Rado Theorem, Perfect matchings, Association scheme, Ratio bound, Clique, Coclique, Quotient graphs, Character table
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