Range of Clique Counts in Graphs

Let $\gamma (G)$ denote the number of cliques in a graph $G$ and let $\mathrm{\Gamma }(n):=\{\gamma (G):\mathrm{\mid }V(G)\mathrm{\mid }=n\}$ be the set of values of $\gamma (G)$ that can be attained on $n$ vertices. We improve on a result by Erdős and Erné to show that $\mathrm{\mid }\mathrm{\Gamma }(n)\mathrm{\mid }\ge 2^{n-4\ln (2){\log }^3(n)}$ for sufficiently large $n$.