In extremal graph theory, there are results such as $$t(C_4,G)\geq t(K_2,G)^4,$$ where $G$ is an undirected graph, $C_4$ is a cycle graph on 4 nodes, $K_2$ is a complete graph of $2$ nodes, and $t(\cdot,G)$ denotes homomorphism density in $G$. Are the similar results for directed graphs, e.g. if $C_4$ is a directed cycle and $G$ is a directed graph?

The homomorphism density $t(F,G)$ of a graph F in G is $\frac{hom(F,G)}{|V(G)|^{|V(F)|}}$, where $hom(F,G)$ is the number of homomorphisms (adjacency-preserving maps) from $V(F)$ to $V(G)$.

homomorphism density? (Or elseeachof ignorant readers like me has to google and such). $\endgroup$ – Włodzimierz Holsztyński Apr 16 '14 at 18:13