All 3-transitive groups satisfy the strict-Erdős–Ko–Rado property

Abstract

A subset $S$ of a transitive permutation group $G\le \mathrm{Sym}\left(n\right)$ is said to be an intersecting set if, for every $g_1,g_2\in S$, there is an $i\in \left[n\right]$ such that $g_1\left(i\right)=g_2\left(i\right)$. The stabilizer of a point in $\left[n\right]$ and its cosets are intersecting sets of size $\left|G\right|/n$. Such families are referred to as canonical intersecting sets. A result by Meagher, Spiga, and Tiep states that if $G$ is a 2-transitive group, then $\left|G\right|/n$ is the size of an intersecting set of maximum size in $G$. In some 2-transitive groups (for instance $\mathrm{Sym}\left(n\right)$, $\mathrm{Alt}\left(n\right)$), every intersecting set of maximum possible size is canonical. A permutation group, in which every intersecting family of maximum possible size is canonical, is said to satisfy the strict-EKR property. In this article, we investigate the structure of intersecting sets in 3-transitive groups. A conjecture by Meagher and Spiga states that all 3-transitive groups satisfy the strict-EKR property. Meagher and Spiga showed that this is true for the 3-transitive group $\mathrm{PGL}(2,q)$. Using the classification of 3-transitive groups and some results in the literature, the conjecture reduces to showing that the 3-transitive group $\mathrm{AGL}(n,2)$ satisfies the strict-EKR property. We show that $\mathrm{AGL}(n,2)$ satisfies the strict-EKR property and as a consequence, we prove Meagher and Spiga’s conjecture. We also prove a stronger result for $\mathrm{AGL}(n,2)$ by showing that “large” intersecting sets in $\mathrm{AGL}(n,2)$ must be a subset of a canonical intersecting set. This phenomenon is called stability.