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 $(\frac 23 +o(1))\binom{n}{4}$. 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 $(0.69 +o(1))\binom{n}{4}$. We also give further evidence that the conjecture is true by proving that the maximum distance between caterpillar trees is at most $(\frac 23 +o(1))\binom{n}{4}$.Comment: arXiv admin note: text overlap with arXiv:1203.272

A problem of Erdős on the minimum number of k-cliques

Fifty years ago Erdős asked to determine the minimum number of k-cliques in a graph on n vertices with independence number less than l. He conjectured that this minimum is achieved by the disjoint union of l − 1 complete graphs of size n l−1. This conjecture was disproved by Nikiforov who showed that the balanced blow-up of a 5-cycle has fewer 4-cliques than the union of 2 complete graphs of size