On the maximum size of minimal definitive quartet sets

TL;DR

This paper explores the maximum size of minimal definitive quartet sets on n leaves, establishing a lower bound of 2n-8, and presents a symmetric construction demonstrating this bound.

Contribution

It provides the first known lower bound for the maximum size of minimal definitive quartet sets and introduces a symmetric construction method.

Findings

Maximum size of minimal definitive quartet sets is at least 2n-8 for n>3

A symmetric construction method is proposed

The problem is shown to be interesting and approachable

Abstract

In this paper, we investigate a problem concerning quartets, which are a particular type 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 n leaves can be, showing that the maximum size is at least 2n-8 for all n>3. This is an enjoyable problem to work on, and we present a pretty construction, which employs symmetry.