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Given a permutation group $G$, the derangement graph of $G$ is the Cayleygraph with connection set the derangements of $G$. The group $G$ is said to beinnately transitive if $G$ has a transitive minimal normal subgroup. Clearly,every primitive group is innately transitive. We show that, besides an infinitefamily of explicit exceptions, there exists a function $f:\mathbb{N}\to\mathbb{N}$ such that, if $G$ is innately transitive of degree $n$ and thederangement 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 typetheorems for permutation groups.

Cliques in derangement graphs for innately transitive groups