On the existence of infinitely many universal tree-based networks.

A tree-based network on a set X of n leaves is said to be universal if any rooted binary phylogenetic tree on X can be its base tree. Francis and Steel showed that there is a universal tree-based network on X in the case of n = 3, and asked whether such a network exists in general. We settle this problem by proving that there are infinitely many universal tree-based networks for any n>1.