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 Erdős and Erné to show that $|\Gamma (n)|\ge 2^{n-4\ln (2){\log }^3(n)}$ for sufficiently large $n$.

9 pages, 1 figure