Subgraphs of weakly quasi-random oriented graphs*

It is an intriguing question to see what kind of information on the structure of an oriented graph D one can obtain if D does not contain a fixed oriented graph H as a subgraph. The related question in the unoriented case has been an active area of research and is relatively well understood in the theory of quasi-random graphs and extremal combinatorics. In this paper, we consider the simplest cases of such a general question for oriented graphs and provide some results on the global behavior of the orientation of D. For the case where H is an oriented four-cycle we prove the following: in every H-free oriented graph D, there is a pair A,B V (D) such that e(A,B) ≥ e(D) ²/32|D|² and e(B,A) ≤ e(A,B)/2. We give a random construction which shows that this bound on e(A,B) is best possible (up to the constant). In addition, we prove a similar result for the case where H is an oriented six-cycle and a more precise result in the case where D is dense and H is arbitrary. We also consider the related extremal question in which no condition is put on the oriented graph D, and we provide an answer that is best possible up to a multiplicative constant. Finally, we raise a number of related questions and conjectures.