Intersecting subsets in finite permutation groups

Let $G\leqslant\mathrm{Sym}(\Omega)$ be transitive, and let $S$ be an intersecting subset, namely, the ratio $xy^{-1}$ of any elements $x,y\in S$ fixes some point. An EKR-type problem is to characterize transitive groups $G\leqslant\mathrm{Sym}(\Omega)$ such that any intersecting set is upper bounded by $|G_\omega|$, where $\omega\in\Omega$. A nice result of Meagher-Spiga-Tiep (2016) tells us that if $G$ is 2-transitive, then indeed $|S|\leqslant|G_\omega|$. A natural next step would be to explore intersecting subsets for primitive groups and quasiprimitive groups. Our study in this paper shows that for quasiprimitive permutation groups, the size $|S|$ can be arbitrarily larger than $|G_\omega|$. We conjecture that for quasiprimitve groups, the upperbound for $|S|$ is $O(|G_\omega||\Omega|^{1\over2})$. As a starting point, we prove that ${|S|/(|G_\omega||\Omega|^{1\over2}})\leqslant{\sqrt2/2}$ for all quasiprimitive actions of the Suzuki groups $G=\mathrm{Sz}(q)$. To show that our conjectured upper bound is tight, we provide examples of groups for which ${|S|/(|G_\omega||\Omega|^{1\over2}})$ is arbitrarily close to ${\sqrt2/2}$. As far as general transitive groups concerned, infinity families of examples produced show that the ratio ${|S|/(|G_\omega||\Omega|^{1\over2}})$ can be arbitrarily large.