Intersection density conjectures for transitive permutation groups

Let $G\leq \operatorname{Sym}(\Omega)$ be a transitive permutation group. For $\omega\in\Omega$, write $G_\omega=\{g\in G:\omega^g=\omega\}$ and define the intersection density of $G$ by

$$\rho(G)=\frac{1}{|G_\omega|}\max\{|\mathcal F|:\mathcal F\subseteq G\text{ is intersecting}\},$$

where $\mathcal F$ is intersecting if for every $g,h\in\mathcal F$ there exists $\omega\in\Omega$ such that $\omega^g=\omega^h$.

Intersection density conjectures. (i) If $|\Omega|$ is a prime power, then $\rho(G)=1$. (ii) If $|\Omega|=2p$, where $p$ is an odd prime, then $1\leq\rho(G)\leq 2$. (iii) If $|\Omega|=pq$, where $p$ and $q$ are two odd primes, then $\rho(G)=1$.

These conjectures extend the Erdős–Ko–Rado theorem to transitive permutation groups and seek sharp restrictions on the size of intersecting families. The third assertion was disproved by constructing transitive groups of degree $pq$, with $p=(q^r-1)/(q-1)$, whose intersection density is $q$; consequently, the combined conjectural statement is refuted.