The number of cliques in graphs of given order and size

Abstract

Let k r ( n , m ) k_{r}\left (n,m\right ) denote the minimum number of r r -cliques in graphs with n n vertices and m m edges. For r = 3 , 4 r=3,4 we give a lower bound on k r ( n , m ) k_{r}\left (n,m\right ) that approximates k r ( n , m ) k_{r}\left (n,m\right ) with an error smaller than n r / ( n 2 − 2 m ) . n^{r}/\left (n^{2}-2m\right ). The solution is based on a constraint minimization of certain multilinear forms. Our proof combines a combinatorial strategy with extensive analytical arguments.