Orbital maximin problem

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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.

Particular cases