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 that H is an oriented four-cycle we prove: in every H-free oriented graph D, there is a pair A, B ⊆ V (D) such that e(A, B) ≥ e(D)/32|D| 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 H is an oriented six-cycle, and a more precise result in the case 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 provide an answer that is best possible up to a multiplicative constant. Finally, we raise a number of related questions and conjectures.