Orbital maximax problem

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Statement

Given a natural number n and a group property \alpha, the orbital maximax problem is the problem of finding a group G satisfying property \alpha acting on a set of size n in such a way that the size of the largest orbital (i.e., orbit under the induced G-action on ordered pairs of distinct elements) is as large as possible.

This is related to the orbital maximin problem, where we try to maximize the size of the smallest orbital.