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.

Particular cases