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 : N → N f\colon\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 ≤ f ⁢ ( k ) n\leq f(k) . Motivation for this work arises from investigations on Erdős–Ko–Rado type theorems for permutation groups.