Erdos conjecture, number of cliques, Turan`s graph
Isn't the conjecture in the paper that the number of $(k+1)$-cliques in this case is at least $c^{k-1}$? That's the number you get by adding a single edge to the Turan-graph. I think it's proved in 1969-10. Also relevant: 1983-28.
Recently, there has been significant progress on bounding the number of cliques as a function of the quadratic term of the number of edges: arxiv:1212.2454.