The intersection densities of transitive actions of ${\mathrm{PSL}}_2(q)$ with cyclic point stabilizers

Given a finite transitive group $G\le \mathrm{Sym}\mathrm{\Omega }$, the {intersection density} of $G$ is defined as the ratio between the size of the largest subsets of $G$ in which any two permutations agree on at least one element of $\mathrm{\Omega }$, and the order of a point stabilizer of $G$. In this paper, we completely determine the intersection densities of the permutation groups ${\mathrm{PSL}}_2(q)$, where $q$ is a power of an odd prime $p$, acting transitively with point stabilizers conjugate to ${\mathbb{Z}}_p$. Our proof uses an auxiliary graph, which is a ${\mathrm{PGL}}_{2}q$-vertex-transitive graph, in which a clique corresponds to an intersecting set of $\text{operaotnrame}{PSL}_2(q)$. For the transitive action of $\text{psl}2q$ with point stabilizers conjugate to ${\mathbb{Z}}_r$, where $r\mid \frac{q-1}{2}$ is an odd prime, we show that the auxiliary graph is not regular, and we construct an intersecting set which is sometimes of maximum size.