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. For r = 3, 4 we give a lower bound on kr (n, m) that approximates kr (n, m) with an error smaller than n r/(n2 - 2m). The solution is based on a constraint minimization of certain multilinear forms. Our proof combines a combinatorial strategy with extensive analytical arguments. © 2010 American Mathematical Society.
Transactions of the American Mathematical Society
Nikiforov, V. (2011). The number of cliques in graphs of given order and size. Transactions of the American Mathematical Society, 363 (3), 1599-1618. https://doi.org/10.1090/S0002-9947-2010-05189-X