Clique numbers of graphs

Abstract

For each natural number n, denote by G(n) the set of all numbers c such that there exists a graph with exactly c cliques (i.e., complete subgraphs) and n vertices. We prove the asymptotic estimate | G( n)| = 0(2 n · n −2/5) and show that all natural numbers between n + 1 and 2 n−6 n 5/6 belong to G( n). Thus we obtain lim n→∞ |G(n)| 2 n =0 , while lim n→∞ |G(n)| a n = ∞ for all 0 < a < 2 .