On the maximum size of minimal definitive quartet sets

In this paper, we investigate a problem concerning quartets; a quartet is a particular kind of tree on four leaves. Loosely speaking, a set of quartets is said to be ‘definitive’ if it completely encapsulates the structure of some larger tree, and ‘minimal’ if it contains no redundant information. Here, we address the question of how large a minimal definitive quartet set on nn leaves can be, showing that the maximum size is at least 2n−82n−8 for all n≥4n≥4. This is an enjoyable problem to work on, and we present a pretty construction, which employs symmetry.