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.

Publication Title

Transactions of the American Mathematical Society