On maximum intersecting sets in direct and wreath product of groups

Abstract

For a permutation group G acting on a set V, a subset I of G is said to be an intersecting set if for every pair of elements g,h∈I there exists v∈V such that g(v)=h(v). The intersection densityρ(G) of a transitive permutation group G is the maximum value of the quotient |I|/|Gv| where Gv is a stabilizer of a point v∈V and I runs over all intersecting sets in G. If Gv is the largest intersecting set in G then G is said to have the Erdős–Ko–Rado (EKR)-property, and moreover, G has the strict-EKR-property if every intersecting set of maximum size in G is a coset of a point stabilizer. Intersecting sets in G coincide with independent sets in the so-called derangement graphΓG, defined as the Cayley graph on G with connection set consisting of all derangements, that is, fixed-point free elements of G. In this paper a conjecture regarding the existence of transitive permutation groups whose derangement graphs are complete multipartite graphs, posed by Meagher, Razafimahatratra and Spiga in Meagher et al. (2021) is proved. The proof uses direct product of groups. Questions regarding maximum intersecting sets in direct and wreath products of groups and the (strict)-EKR-property of these group products are also investigated. In addition, some errors appearing in the literature on this topic are corrected.