Counting <i>H</i>-free orientations of graphs

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