Cliques in derangement graphs for innately transitive groups

Abstract: Given a permutation group $G$, the derangement graph of $G$ is the Cayley graph with connection set the derangements of $G$. The group $G$ is said to be innately transitive if $G$ has a transitive minimal normal subgroup. Clearly, every primitive group is innately transitive. We show that, besides an infinite family of explicit exceptions, there exists a function $f:\mathbb{N}\to \mathbb{N}$ such that, if $G$ is innately transitive of degree $n$ and the derangement graph of $G$ has no clique of size $k$, then $n\le f(k)$. Motivation for this work arises from investigations on Erd\H{o}s-Ko-Rado type theorems for permutation groups.