# Separating subsets from their images

## TL;DR

This paper introduces a new parameter for transitive permutation groups, studies its bounds and behavior, and classifies groups with the maximum possible value of this parameter.

## Contribution

It defines the parameter ${f m}(G)$, provides bounds and asymptotic analysis for transitive groups, and classifies those with the largest ${f m}(G)$ value.

## Key findings

- Derived bounds for ${f m}(G)$ in transitive groups.
- Analyzed asymptotic behavior in primitive groups.
- Classified groups with maximal ${f m}(G)$.

## Abstract

Let $G$ be a transitive permutation group acting on $\Omega$. In this paper, we introduce and study the parameter ${\bf m}(G)$, 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 ${\bf m}(G)$ largest possible, namely with ${\bf m}(G)=\lceil (|\Omega|+1) / 2 \rceil$.