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))(n4). 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))(n4). 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))(n4).

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