A generalization of Turán's theorem to directed graphs

We consider an extremal problem for directed graphs which is closely related to Turán's theorem giving the maximum number of edges in a graph on n vertices which does not contain a complete subgraph on m vertices. For an integer n?2, let T_n denote the transitive tournament with vertex set X_n={1,2,3,…,n} and edge set {(i,j):1?i<j?n}. A subgraph H of T_n is said to be m-locally unipathic when the restriction of H to each m element subset of X_n consisting of m consecutive integers is unipathic. We show that the maximum number of edges in a m-locally unipathic subgraph of T_n is ($\text{q}\text{2}\text{)(m?1)}{}^2\text{+q(m?1)r+?}\text{1}\text{4}\text{r}{}^2\text{?}$ where n= q(m?1+r and $\text{?}\text{1}\text{2}\text{(m?1)??r<?}\text{3}\text{2}\text{(m?1)?}$. As is the case with Turán's theorem, the extremal graphs for our problem are complete multipartite graphs. Unlike Turán's theorem, the part sizes will not be uniform. The proof of our principal theorem rests on a combinatorial theory originally developed to investigate the rank of partially ordered sets.