This talk concerns new results pertaining to the following question. Assuming that we restrict the set of phylogenetic networks to some subclass, what is the maximum value of 0 ≤ p ≤ 1 such that for every input set T of rooted triplets, there exists some network N(T) from the subclass such that at least p|T| of the triplets are consistent with N(T)?

Joint work with Jaroslaw Byrka and Katharina T. Huber.