Counting $H$-free orientations of graphs

In 1974, Erd\H{o}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 Erd\H{o}s's question. Moreover, we determine the answer exactly when $H$ is an odd cycle and $n$ is sufficiently large, answering a question of Ara\'ujo, Botler and Mota.