Separating subsets from their images

Abstract Let G be a transitive permutation group acting on Ω $\Omega $ normal upper Omega . In this paper, we introduce and study the parameter sep ( G ) $\mathrm {sep}(G)$ sep left parenthesis upper G right parenthesis , which denotes the size of the smallest set of points A such that, for every permutation g ∈ G $g\in G$ g element of upper G , A ∩ A g $A \cap A^g$ upper A intersection upper A Superscript g is nonempty. In particular, we focus on deriving general bounds for arbitrary transitive groups, and on the asymptotic behaviour of certain families of primitive groups. We also provide a classification of transitive groups with sep ( G ) $\mathrm {sep}(G)$ sep left parenthesis upper G right parenthesis largest possible, namely with sep ( G ) = ⌈ ( | Ω | + 1 ) / 2 ⌉ $\mathrm {sep}(G)=\lceil (|\Omega |+1) / 2 \rceil $ sep left parenthesis upper G right parenthesis equals left ceiling left parenthesis StartAbsoluteValue normal upper Omega EndAbsoluteValue plus 1 right parenthesis divided by 2 right ceiling .