Thinking of an edge as of a 2$2$-clique, it's natural to consider a slightly more general question than Turan considered in his celebrated theorem: given r \le k \le n$r \le k \le n$, what is the maximal possible number of r$r$-cliques in a graph X$X$ on n$n$ vertices without k$k$-cliques?. Has this question been considered in the literature?
Yet more generally, given two graphs G$G$ and H$H$ and a number n$n$, one could ask for the maximum of the homomorphism number hom(G,X)$hom(G,X)$ over all graphs X$X$ on n$n$ vertices satisfying hom(H,X)=0$hom(H,X)=0$. Any references for this one maybe?
