GENERALIZED TURAN’S GRAPH THEOREM - Khadzhiivanov, Nikolay; Nenov, Nedyalko (Annual of Sofia University St. Kliment Ohridski. Faculty of Mathematics and Informatics, 2004)
Let $G$ be an $n$-vertex graph and there is a vertex of $G$ which is contained in maximum number of $p$-cliques, but is not contained in $(s+1)$-clique, where ${2\le p\le\min(s,n)}$. Then the number of $p$-cliques of $G$ is less than the number of $p$-cliques in the $n$-vertex $S$-partite Tur\'an's graph $T_s(n)$ or $G=T_s(n)$.