Number of graphs with M edges that does not contain K-clique

Question

If we consider the space of graphs $G(n,M)$ with $n$ vertices and $M$ denotes the number of edges. Is there any way of upper bounding the number of graphs in this space that does not contain any k-cliques? Can it perhaps be done in some way using the closely related space $G(n,p)$ with edge probability $p$?

Answer 1

The asymptotics are determined in this paper of Erdos, Frankl, and Rodl: http://link.springer.com/article/10.1007%2FBF01788085

Let $H$ be a fixed graph of chromatic number $r$. It is shown that the number of graphs on $n$ vertices and not containing $H$ as a subgraph is $2^{{n\choose 2}(1−\frac{1}{r−1}+o(1))}$.

That is, the number is about what you'd expect from fixing a Turan graph and taking all its possible subgraphs.