Orientation Ramsey thresholds for cycles and cliques

If $G$ is a graph and $vec H$ is an oriented graph, we write $Gto vec H$ to say that every orientation of the edges of $G$ contains $vec H$ as a subdigraph. We consider the case in which $G=G(n,p)$, the binomial random graph. We determine the threshold $p_{vec H}=p_{vec H}(n)$ for the property $G(n,p)to vec H$ for the cases in which $vec H$ is an acyclic orientation of a complete graph or of a cycle.