The exact bound for the Erdős-Ko-Rado theorem for $t$-cycle-intersecting permutations

Abstract

In this paper we adapt techniques used by Ahlswede and Khachatrian in their proof of the Complete Erdős-Ko-Rado Theorem to show that if $n\ge 2t+1$, 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.

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