All \(3\)-transitive groups satisfy the strict-EKR property

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.