  Let k_r(n,m) denote the minimum number of r-cliques in graphs with n vertices
and m edges. For r=3,4 we give a lower bound on k_r(n,m) that approximates
k_r(n,m) with an error smaller than n^r/(n^2-2m). The solution is based on a
constraint minimization of certain multilinear forms. In our proof, a
combinatorial strategy is coupled with extensive analytical arguments.
