Fix integers $V$ and $E_{\text{max}}$, and consider graphs $G$ with $V$ vertices and at most $E_{\text{max}}$ edges. What is the best lower bound that one can give on the number of distinct independent sets that exist in $G$?

(You can also phrase this in terms of cliques; we're asking if a graph with a large enough number of edges is forced to have a lot of cliques.)

Note that there are many results, proceeding from the classical work of Turan, giving good lower bounds for the number of $k$-cliques for some fixed $k$, and this of course gives us *some* bounds on the number of cliques overall. But I'm interested in the best bounds one can find on the number of cliques overall, and this seems to be a different problem.

This paper of Wood solves the opposite problem (maximizing the number of cliques for fixed $V$ and $E$). Wood's paper also says 'lower bounds on the number of cliques in a graph have also been obtained' and gives a number of references. But those references all seem to concern Turan type problems where we try to minimize the number of $k$ cliques for some fixed $k$, not the total number of cliques overall.