The number of cliques in graphs of given order and size

Let kr (n, m) denote the minimum number of r-cliques in graphs with n vertices and m edges. We give a lower bound on kr (n, m) that approximates kr (n, m) with an error smaller than n r / n 2 − 2m � . This essentially solves a sixty year old problem. The solution is based on a constraint minimization of certain multilinear forms. In our proof, a combinatorial strategy is coupled with extensive analytical arguments.