  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.
