On the maximum size of 2-weakly compatible split systems

Abstract: We consider a Turán-type problem arising in phylogenetics: determining the maximum size of a 2-weakly compatible split system. This compatibility condition arises in the reconstruction of phylogenetic networks from quartet weights. It was previously shown that a 2-weakly compatible split system has size at most [ 3\binom{n}{4}+\binom{n}{2}. ] We prove that the maximum size is $O(n<sup>{5/2})$.