Frankl's diversity theorem for permutations

TL;DR

This paper extends Frankl's diversity theorem, originally for set families, to permutations, providing a stability result that relates the size of permutation families to their proximity to trivial families, for large n.

Contribution

It establishes a permutation analogue of Frankl's diversity theorem, extending stability results to permutation groups for sufficiently large n.

Findings

Proves a Frankl-type stability theorem for permutations

Extends Hilton--Milner type results to permutation families

Provides bounds on permutation family sizes based on their structure

Abstract

In 1987, Frankl proved an influential stability result for the Erd\H os--Ko--Rado theorem, which bounds the size of an intersecting family in terms of its distance from the nearest (subset of) star or trivial intersecting family. It is a far-reaching extension of the Hilton--Milner theorem. In this paper, we prove its analogue for permutations on $\{1,\ldots, n\}$, provided $n$ is large. This provides a similar extension of a Hilton--Milner type result for permutations proved by Ellis.