The exact bound for the Erd\H{o}s-Ko-Rado theorem for $t$-cycle-intersecting permutations

Karen Meagher; Alison Purdy

The exact bound for the Erd\H{o}s-Ko-Rado theorem for $t$-cycle-intersecting permutations

Combinatorics 2013-03-05 v2

Authors: Karen Meagher , Alison Purdy

Abstract

In this paper we adapt techniques used by Ahlswede and Khachatrian in their proof of the Complete Erd\H{o}s-Ko-Rado Theorem to show that if $n \geq 2t+1$ , then any pairwise $t$ -cycle-intersecting family of permutations has cardinality less than or equal to $(n-t)!$ . Furthermore, the only families attaining this size are the stabilizers of $t$ points, that is, families consisting of all permutations having $t$ 1-cycles in common. This is a strengthening of a previous result of Ku and Renshaw and supports a recent conjecture by Ellis, Friedgut and Pilpel concerning the corresponding bound for $t$ -intersecting families of permutations.

Keywords

erd\H{o}s-ko-rado theorem permutations t-cycle-intersecting stabilizers extremal set theory combinatorics

Cite

@article{arxiv.1208.3638,
  title  = {The exact bound for the Erd\H{o}s-Ko-Rado theorem for $t$-cycle-intersecting permutations},
  author = {Karen Meagher and Alison Purdy},
  journal= {arXiv preprint arXiv:1208.3638},
  year   = {2013}
}

Comments

23 pages; article unchanged; After v1 posted, the authors were made aware of a 2011 paper by V.M. Blinovsky which uses a similar method to give the size of the largest family for all n and t. Our article may still be of interest for its explicit characterization of the largest families, its use of generating sets and the additional background information and references

The exact bound for the Erd\H{o}s-Ko-Rado theorem for $t$-cycle-intersecting permutations

Abstract

Keywords

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