Frankl's diversity theorem for permutations

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 $\{1,\dots ,n\}$, provided $n$ is large. This provides a similar extension of a Hilton–Milner type result for permutations proved by Ellis.