On the maximum quartet distance between phylogenetic trees

A conjecture of Bandelt and Dress states that the maximum quartet distance between any two phylogenetic trees on n leaves is at most ( 2/3 +o(1))(n 4). Using the machinery of flag algebras, we improve the currently known bounds regarding this conjecture; in particular, we show that the maximum is at most (0.69 + o(1)) (n 4). We also give further evidence that the conjecture is true by proving that the maximum distance between caterpillar trees is at most ( 2/3 + o(1)) (n 4).