Extremal digraph results for topological complete subgraphs (Q1272769)

The author shows that if a digraph \(D_n\) has at least \(n(n-1)/2+44p^2n\) edges, then \(D_n\) contains a topological complete digraph with \(p\) vertices; but that this conclusion no longer holds if \(D_n\) has only \(n(n-1)/2+p^2n/16\) edges. He also obtains an analogous result on containing topological tournaments and a Turán-type result for containing a topological transitive tournament and a transitive tournament.