All $3$-transitive groups satisfy the strict-EKR property

Abstract

A subset $S$ of a transitive permutation group $G \leq \mathrm{Sym}(n)$ is said to be an intersecting set if, for every $g_{1},g_{2}\in S$, there is an $i \in [n]$ such that $g_{1}(i)=g_{2}(i)$. The stabilizer of a point in $[n]$ and its cosets are intersecting sets of size $|G|/n$. 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 $|G|/n$ is the size of an intersecting set of maximum size in $G$. In some $2$-transitive groups (for instance $\mathrm{Sym}(n)$, $\mathrm{Alt}(n)$), 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 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.