Separating subsets from their images

Barbieri, Marco (author), Lekše, Maruša (author), Potočnik, Primož (author), Rekvényi, Kamilla (author)

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

• Language: English
• Keywords: permutation groups, transitive groups, primitive groups, self-separable set
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Year: 2026
• Number of pages: 47 str.
• Numbering: Vol. 14, art. e111
• PID: 20.500.12556/RUL-185118
• UDC: 512:519.1
• ISSN on article: 2050-5094
• DOI: 10.1017/fms.2026.10258
• COBISS.SI-ID: 285574915
• Publication date in RUL: 23.07.2026

Record is a part of a journal

• Title: Forum of mathematics : Sigma
• Shortened title: Forum math. Sigma
• Publisher: Cambridge University Press
• ISSN: 2050-5094
• COBISS.SI-ID: 522596121

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.