Using the existence of t-designs to prove Erdős–Ko–Rado

Godsil, Chris; Guo, Krystal

doi:10.1016/j.disc.2018.11.006

Cited by 2 publications

(6 citation statements)

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“…This approach only works for some values of k. The existence of such a clique implies the existence of a weighted adjacency matrix M of M 2 (2k), for which the Delsarte-Hoffman bound holds with equality. This is similar to the result in [6], where the authors show how to find the matrix used in Wilson's proof of the EKR Theorem [20]. This is the motivation for our second approach, where we show that such a matrix exists, even when there is no appropriate clique.…”

Section: Clique-coclique Approachsupporting

confidence: 82%

“…In 2015, Godsil and Meagher applied this method to the family of all perfect matchings of the complete graph K 2k to find the largest set of intersecting perfect matchings (t = 1) [8]; later in 2017 it was applied to t-intersecting perfect matchings by Lindzey [12]. (6)). In Definition 2.2, let t = 1 and 2k = 6.…”

Section: Definition 22 ([8]mentioning

confidence: 99%

“…Two vertices here are adjacent if they are not intersecting. The graph depicted in Figure 1 below is M 1 (6). 2.2.…”

Section: Definition 22 ([8]mentioning

confidence: 99%

“…The Gurobi Optimizer [10] is then used to find solutions for these system of inequalities. As such we determined the desired weighted adjacency matrices as follows: A [6,6,4,2] .…”

Section: Algebraic Combinatorics Vol 4 #4 (2021)mentioning

confidence: 99%

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

Fallat¹,

Meagher²,

Shirazi³

2021

Algebraic Combinatorics

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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

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Section: Clique-coclique Approachsupporting

confidence: 82%

Section: Definition 22 ([8]mentioning

confidence: 99%

“…Two vertices here are adjacent if they are not intersecting. The graph depicted in Figure 1 below is M 1 (6). 2.2.…”

Section: Definition 22 ([8]mentioning

confidence: 99%

“…The Gurobi Optimizer [10] is then used to find solutions for these system of inequalities. As such we determined the desired weighted adjacency matrices as follows: A [6,6,4,2] .…”

Section: Algebraic Combinatorics Vol 4 #4 (2021)mentioning

confidence: 99%

See 2 more Smart Citations