Range of Clique Counts in Graphs

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