On the EKR Module property

TL;DR

This paper explores the EKR-module property in permutation groups, presenting multiple infinite families that satisfy this property, thereby highlighting its diversity among permutation groups.

Contribution

It introduces several infinite families of permutation groups that satisfy the EKR-module property, expanding understanding of its prevalence.

Findings

Multiple infinite families satisfy the EKR-module property

Permutation groups with this property are diverse

The property extends the classical EKR theorem to broader contexts

Abstract

In the recent years, the generalization of the Erd\H{o}s-Ko-Rado (EKR) theorem to permutation groups has been of much interest. A transitive group is said to satisfy the EKR-module property if the characteristic vector of every maximum intersecting set is a linear combination of the characteristic vectors of cosets of stabilizers of points. This generalization of the well-know permutation group version of the Erd\H{o}s-Ko-Rado (EKR) theorem, was introduced by K. Meagher. In this article, we present several infinite families of permutation groups satisfying the EKR-module property, which shows that permutation groups satisfying this property are quite diverse.