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

TL;DR

This paper establishes the exact maximum size of $t$-cycle-intersecting permutation families for $n \\geq 2t+1$, confirming that stabilizers of $t$ points are extremal, thus strengthening previous bounds and supporting a conjecture.

Contribution

It proves the exact bound for $t$-cycle-intersecting permutations and characterizes the extremal families, extending prior results and confirming a recent conjecture.

Findings

Maximum size of $t$-cycle-intersecting families is $(n-t)!$ for $n \\geq 2t+1$

Extremal families are stabilizers of $t$ points

Supports the conjecture by Ellis, Friedgut, and Pilpel

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.