Erd\H{o}s-Ko-Rado problems for permutation groups

Abstract

In this paper, we study intersecting sets in primitive and quasiprimitive permutation groups. Let $G \leqslant \mathrm{Sym}(\Omega)$ be a transitive permutation group, and ${S}$ an intersecting set. Previous results show that if $G$ is either 2-transitive or a Frobenius group, then $|{S}|\leqslant|G_{\omega}|$ (for some $\omega \in \Omega$). Furthermore, for some 2-transitive groups, $|{S}|=|G_{\omega}|$ if and only if ${S}$ is a coset of a stabilizer. In this paper, we prove that these statements are far from the truth for general transitive groups. In particular, we show that in the case of primitive groups, there is even no absolute constant $c$ such that $|{S}|\leqslant c|G_\omega|$. In the case $G$ is a primitive permutation group isomorphic to $\mathrm{PSL(2,p)}$, we characterize the subgroups of $G$ which are intersecting sets. We also show that if $G \leqslant \mathrm{Sym}(\Omega)$ is a permutation group of prime power degree, then for any intersecting set $S$, we have $|S|\leq |G_{\omega}|$ (for some $\omega \in \Omega$). This proves a part of a conjecture in \cite{MRS}.