Separating subsets from their images

Let $G$ be a transitive permutation group acting on $\mathrm{\Omega }$. In this paper, we introduce and study the parameter $\mathbf{m}(G)$, which denotes the size of the smallest set of points $A$ such that, for every permutation $g\in G$, $A\cap A^g$ 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 $\mathbf{m}(G)$ largest possible, namely with $\mathbf{m}(G)=\lceil (\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }+1)\mathrm{/}2\rceil$.