On the Erdos-Ko-Rado property for finite Groups

2013/10/06 by Mohammad Bardestani, Bardestani, Mohammad, Keivan Mallahi-Karai +1
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #Limits and Structures in Graph Theory #math.CO #math.GR

paper · pdf · doi:10.48550/arxiv.1310.1643

This is the final version. To appear in the Journal of Algebraic Combinatorics

openalex publication_date 2013/10/06 · arxiv created 2014/12/12 · arxiv updated 2014/12/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let a finite group G act transitively on a finite set X. A subset S⊆ G is said to be \it intersecting if for any s₁,s₂∈ S, the element s₁^-1s₂ has a fixed point. The action is said to have the \it weak Erdős-Ko-Rado property, if the cardinality of any intersecting set is at most |G|/|X|. If, moreover, any maximal intersecting set is a coset of a point stabilizer, the action is said to have the \it strong Erdős-Ko-Rado property. In this paper we will investigate the weak and strong Erdős-Ko-Rado property and attempt to classify the groups whose all transitive actions have these properties. In particular, we show that a group with the weak Erdős-Ko-Rado property is solvable and that a nilpotent group with the strong Erdős-Ko-Rado property is product of a 2-group and an abelian group of odd order.

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