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.

翻译：1987年，弗兰克证明了关于埃尔德什-柯-拉多定理的一个具有影响力的稳定性结果，该结果通过相交族到最近星族或平凡相交族（的子集）的距离来界定其规模。这是希尔顿-米尔纳定理的一个深远推广。本文证明了该结果在集合 $\{1,\ldots, n\}$ 上置换情形的类比，其中 $n$ 充分大。这为埃利斯所证明的置换情形的希尔顿-米尔纳型结果提供了类似的推广。