2023/11/21 by Pantangi, Venkata Raghu Tej
#05D99 #05E18 #05E30 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2311.13055
A subset S of a transitive permutation group G ≤ Sym(n) is said to be an intersecting set if, for every g1,g2∈ S, there is an i ∈ [n] such that g1(i)=g2(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 Sym(n), 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 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 AGL(n,2) satisfies the strict-EKR property. We show that 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 AGL(n,2) by showing that "large" intersecting sets in AGL(n,2) must be a subset of a canonical intersecting set. This phenomenon is called stability.