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.