# Orbital maximin problem

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.