A comment on Intersecting Families of Permutations

TL;DR

This paper critiques a proof regarding the maximum size of t-intersecting permutation families, clarifying that the original characterization of extremal families relies on prior work by Ellis, despite the flawed proof.

Contribution

The paper identifies an error in the original proof and clarifies that the extremal characterization follows from existing results by Ellis.

Findings

The original proof of the extremal characterization is incorrect.

The extremal families are characterized by Ellis's earlier work.

The maximum size of t-intersecting permutation families is (n-t)!.

Abstract

Ellis, Friedgut and Pilpel proved that for large enough $n$, a $t$-intersecting family of permutations contains at most $(n-t)!$ permutations. Their main theorem also states that equality holds only for $t$-cosets. We show that their proof of the characterization of extremal families is wrong. However, the characterization follows from a paper of Ellis, as mentioned already by Ellis, Friedgut and Pilpel.