Orbital maximin problem

Statement
For a given natural number $$n$$ and a group property $$\alpha$$, the orbital maximin problem for the pair $$(n,\alpha)$$ is the problem of finding a group $$G$$ satisfying $$\alpha$$ with an action on a set of size $$n$$ such that the minimum of the sizes of the orbitals under the action of $$G$$ is as large as possible.

Here, an orbital is an orbit under the induced action of $$G$$ on unordered pairs of distinct elements from the set.

In general, we take $$\alpha$$ to be a property that is both subgroup-closed and quotient-closed. For such $$\alpha$$, we can restrict attention to faithful group actions.

This is related to the orbital maximax problem, where we try to maximize the size of the largest orbital.