Let $\Gamma$ be a connected $G$-vertex-transitive graph, let $v$ be a vertex
of $\Gamma$ and let $L=G_v^{\Gamma(v)}$ be the permutation group induced by the
action of the vertex-stabiliser $G_v$ on the neighbourhood $\Gamma(v)$. Then
$(\Gamma,G)$ is said to be \emph{locally-$L$}. A transitive permutation group
$L$ is \emph{graph-restrictive} if there exists a constant $c(L)$ such that,
for every locally-$L$ pair $(\Gamma,G)$ and an arc $(u,v)$ of $\Gamma$, the
inequality $|G_{uv}|\leq c(L)$ holds.

Let G=PGL(2,q) be the projective general linear group acting on the
projective line P_q. A subset S of G is intersecting if for any pair of
permutations \pi,\sigma in S, there is a projective point p in P_q such that
p^\pi=p^\sigma. We prove that if S is intersecting, then the size of S is no
more than q(q-1). Also, we prove that the only sets S that meet this bound are
the cosets of the stabilizer of a point of P_q.