On the intersection density of the Kneser Graph $K(n,3)$

TL;DR

This paper investigates the intersection density of the Kneser graph $K(n,3)$, determining it for cases where the automorphism group contains $ ext{PSL}_2(q)$, and explores related properties for $K(n,2)$.

Contribution

It provides the first explicit determination of the intersection density of $K(n,3)$ under certain automorphism group conditions, expanding understanding of intersecting sets in Kneser graphs.

Findings

Intersection density of $K(n,3)$ is determined when automorphism group contains $ ext{PSL}_2(q)$.

Explicit results depend on the congruence class of $q$.

Brief analysis of intersection density for $K(n,2)$ with $ ext{PSL}_2(q)$ subgroup presence.

Abstract

A set $\mathcal{F} \subset \operatorname{Sym}(V)$ is \textsl{intersecting} if any two of its elements agree on some element of $V$. Given a finite transitive permutation group $G\leq \operatorname{Sym}(V)$, the \textsl{intersection density} $\rho(G)$ is the maximum ratio $\frac{|\mathcal{F}||V|}{|G|}$ where $\mathcal{F}$ runs through all intersecting sets of $G$. The \textsl{intersection density} $\rho(X)$ of a vertex-transitive graph $X = (V,E)$ is equal to $\max \left\{ \rho(G) : G \leq \operatorname{Aut}(X), \mbox{ $G$ transitive} \right\}$. In this paper, we study the intersection density of the Kneser graph $K(n,3)$, for $n\geq 7$. The intersection density of $K(n,3)$ is determined whenever its automorphism group contains $\operatorname{PSL}_{2}(q)$, with some exceptional cases depending on the congruence of $q$. We also briefly consider the intersection density of $K(n,2)$ for values of $n$ where $\operatorname{PSL}_{2}(q)$ is a subgroup of its automorphism group.