Range of Clique Counts in Graphs

Abstract

Let $\gamma(G)$ denote the number of cliques in a graph $G$ and let $\Gamma(n):= \{\gamma(G):|V(G)|=n\}$ be the set of values of $\gamma(G)$ that can be attained on $n$ vertices. We improve on a result by Erd\H{o}s and Ern\'e to show that $| \Gamma(n)| \geq 2^{n-4\ln(2)\log^3(n)}$ for sufficiently large $n$.