It is well known that ex(n;C3) = (n2)/4 (maximal number of arcs in a graph on n vertices without cycle of length 3), and the extremal graph is the complete bipartite graph. For excluded 4-cycle graphs, the exact value of ex(n;C_4) is known for all values of n of the form n = q^2 + q + 1, where q is a power of 2, or a prime power exceeding 13. In this talk we recall how to drive an easy bound of type O(n^(3/2)) and the aim is to study in some sense analogue questions for directed case. Since acyclic digraphs can have a lot of arcs, conditioning only on the number of arcs can not garanty anything about the existence of directed cycles. On the other hand, if a graph has a lot of k-cycles, taking a random orientation should provide a directed k-cycle. This leads us to define a new notion of pseudo-randomness for oriented graphs. Oriented graphs having this pseudo-randomness property are what we call mixed digraphs. For this family of digraphs, we prove that a simple condition on the number of arcs can garanty the existence of 4-cycles.
Joint work with Omid Amini and Simon Griffith.