On the maximum quartet distance between phylogenetic trees (Q2800184)

Consider two phylogenetic trees \(T_1\) and \(T_2\) on \(n\) leaves; the quartet distance is defined as the difference between \({n\choose4}\) and the number of quartets compatible to both \(T_1\) and \(T_2\). Using flag algebras, the authors show that the maximum quartet distance between two phylogenetic trees on \(n\) leaves is at most \((0.69+o(1)){n\choose4}\), close to a conjecture of \textit{H.-J. Bandelt} and \textit{A. Dress} [Adv. Appl. Math. 7, 309--343 (1986; Zbl 0613.62083)] supposing the upper bound as \((2/3+o(1)){n\choose4}\). Moreover, it is shown that the maximum quartet distance between two phylogenetic caterpillar trees on \(n\) leaves is at most \((2/3+o(1)){n\choose4}\).