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?