Cliques in derangement graphs for innately transitive groups

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 W N ! 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 ≤ f .k/. Motivation for this work arises from investigations on Erdos–Ko–Rado type theorems for permutation groups.