No Hilton-Milner type results for linear groups of degree two

Abstract

A set of permutations F of a finite transitive permutation group G ≤ Sym(Ω) is intersecting if any pair of elements of F agree on an element of Ω. We say that G has the EKR property if an intersecting set of G has size at most the order of a point stabilizer. Moreover, G has the strict-EKR property whenever G has the EKR property and any intersecting set of maximum size is a coset of a point stabilizer of G.It is known that the permutation group GL2(Fq) acting on Ωq := F 2 q \ {0} has the EKR property, but does not have the strict-EKR property since the stabilizer of a hyperplane is a maximum intersecting set. In this paper, it is proved that the Hilton-Milner type result does not hold for GL2(Fq) acting on Ωq. Precisely, it is shown that a maximal intersecting set of GL2(Fq) is of maximum size. As a result, we prove the Complete Erdős-Ko-Rado theorem for GL2(Fq).

Stability of Erdős-Ko-Rado Theorems in Circle Geometries

“…Thus, proving a Hilton-Milner type result for circle geometries could have interesting consequences for the size of the largest 2-intersecting families. Recently, Maleki and Razafimahatratra [MR21] proved that the only intersecting families in GL(2, q) acting on F 2 q \ {0} are stabilisers of a point or a hyperplane. This is in contrast to the existence of Hilton-Milner type families in PGL(2, q) acting on PG(1, q), or equivalently in the ovoidal Minkowski planes.…”