Counting H-free orientations of graphs

In 1974, ErdÅ's posed the following problem. Given an oriented graph H, determine or estimate the maximum possible number of H-free orientations of an n-vertex graph. When H is a tournament, the answer was determined precisely for sufficiently large n by Alon and Yuster. In general, when the underlying undirected graph of H contains a cycle, one can obtain accurate bounds by combining an observation of Kozma and Moran with celebrated results on the number of F-free graphs. As the main contribution of the paper, we resolve all remaining cases in an asymptotic sense, thereby giving a rather complete answer to ErdÅ's's question. Moreover, we determine the answer exactly when H is an odd cycle and n is sufficiently large, answering a question of Araújo, Botler and Mota.