Denote by $\omega(G)$ the size of a maximum clique of a graph $G$.
Suppose $G$ is a $K_r$-free graph (i.e. $\omega(G) < r$), and let $G'$ be a graph obtained by adding at most $k$ edges to $G$.
What is the maximum possible value of $\omega(G')$ ?
I'm obviously interested in a general answer in terms of $r$ and $k$, and which graphs attain this maximum, but any partial results / bounds for specific or general values. Also references, if they exist.
If it helps, one other way to view this question is : for some $r' > r$, what is the minimum number of edges that are necessary to add to a $K_r$-free graph $G$ so that $\omega(G') = r'$ ?
If you add any edge that does not belong to the first $K_{r'}$ you create, your actions are sub-optimal. So you can restrict yourself to $K_{r'}$ as universe and ask for the smallest number of edges you need to remove to create a graph that is $K_{r+1}$-free.
The answer is given by Turán's theorem (https://en.wikipedia.org/wiki/Tur%C3%A1n%27s_theorem).
(I may have reinterpreted your $r$ and/or $r'$, but you probably get the essential part).