EKR-Module Property

Let $G$ be a finite group acting transitively on $X$. We say $g,h \in G$ are intersecting if $gh^{-1}$ fixes a point in $X$. A subset $S$ of $G$ is said to be an intersecting set if every pair of elements in $S$ intersect. Cosets of point stabilizers are canonical examples of intersecting sets. The group action version of the classical Erdos-Ko-Rado problem asks about the size and characterization of intersecting sets of maximum possible size. A group action is said to satisfy the EKR property if the size of every intersecting set is bounded above by the size of a point stabilizer. A group action is said to satisfy the strict-EKR property if every maximum intersecting set is a coset of a point stabilizer. It is an active line of research to find group actions satisfying these properties. It was shown that all $2$-transitive satisfy the EKR property. While some $2$-transitive groups satisfy the strict-EKR property, not all of them do. However a recent result shows that all $2$-transitive groups satisfy the slightly weaker "EKR-module property"(EKRM), that is, the characteristic vector of a maximum intersecting set is a linear span of characteristic vectors of cosets of point stabilizers. We will discuss about a few more infinite classes of group actions that satisfy the EKRM property. I will also provide a few non-examples and a characterization of the EKRM property using characters of $G$ .