Separating subsets from their images

TL;DR

The paper introduces and studies the parameter ${f m}(G)$, the size of the smallest non-self-separable subset for a transitive permutation group $G$, tying it to a graph-packing viewpoint via self-separability. It establishes general bounds, derives monotonicity properties, and reduces many questions to primitive groups, where detailed analysis across the eight O'Nan–Scott classes yields asymptotic bounds and explicit classifications. The authors prove Neumann’s separation lemma for the lower bound, connect regular groups to difference-bases, and obtain sharp upper-bound attainments in both primitive and imprimitive cases, culminating in a near-complete classification of groups achieving ${f m}(G)=igl earrow (n+1)/2igr earrow$ in Section 7. They then specialize to primitive groups, providing asymptotic estimates for almost simple and diagonal types in their standard actions and outlining implications for broader families, with implications for the structure of permutation groups and incidence-geometric interpretations. Overall, the work links self-separability with core permutation-group structure, providing precise bounds, constructive examples, and a framework for further asymptotic and computational exploration.

Abstract

Let $G$ be a transitive permutation group acting on $Ω$. 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 ${\bf m}(G)=\lceil (|Ω|+1) / 2 \rceil$.

Theorems & Definitions (72)

• Definition 1
• Remark 3
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Definition 5
• Corollary 7
• Corollary 8
• Theorem 9