Given a group $G$ acting transitively on a set $X$ of $n$ points, consider the induced action on the set $\binom{X}{k}$ of $k$-element subsets of $X$. Obviously, if $k>n/2$, the orbit of any set is *intersecting*, i.e., every two sets in the orbit intersect. How much better than $n/2$ can we do?

For an example to see what I am talking about, considering the action of $\text{PSL}_3(\mathbf{F}_p)$ on lines in $\mathbf{P}^2(\mathbf{F}_p)$ gives an example with $k\approx \sqrt{n}$. Is $k\approx n^{1/3}$ possible? What about $k\approx \log n$?

I am interested in asymptotics, so take $n$ to be large but otherwise however you like. You may also take $G$ to be any transitive subgroup of $S_n$.