Is there any result on the converse of Turan's Theorem?

Question

I can't seem to find any information on this, but I was working on a graph theory problem when I came across a situation where adding any $k$ edges would induce a clique of size $r$ and I was trying to deduce a minimum number of edges. I know Turan's Theorem says that if you have more than $t_{n,r}$ edges on a $n$-vertex graph than you have the clique $K_r$, but I didn't know if there was any result saying that if adding any $k$ edges induces a clique of size $r$ then you have at least some number of edges.