Thinking of an edge as of a $2$-clique, it's natural to consider a slightly more general question than Turan considered in his celebrated theorem: **given $r \le k \le n$, what is the maximal possible number of $r$-cliques in a graph $X$ on $n$ vertices without $k$-cliques?** Has this question been considered in the literature?

Yet more generally, given two graphs $G$ and $H$ and a number $n$, one could ask for the **maximum of the homomorphism number $hom(G,X)$ over all graphs $X$ on $n$ vertices satisfying $hom(H,X)=0$.** Any references for this one maybe?