Summary: Let \(G\) be a graph with a clique number \(w\). For \(1\leq j\leq w\), let \(k_ j\) be the number of complete subgraphs on \(j\) nodes. We show that \(k_{j+1} \leq(\begin{smallmatrix} w\\ j+1\end{smallmatrix})(k_ j/(\begin{smallmatrix} w\\ j\end{smallmatrix}))^{(j+ 1)/j}\). This is exact for complete balanced \(w\)-partite graphs and gives Turán’s theorem when \(j= 1\). A corollary is \[ w\geq {8k^ 3_ 2- 9k^ 2_ 3+ 3k_ 3 \sqrt{16k^ 3_ 2+ 9k^ 2_ 3}\over 4k^ 3_ 2- 18k^ 2_ 3}. \] This new bound on the clique number supersedes an earlier bound from Turán’s theorem.