Consider all graphs with E many edges. The question is to find the maximum number of triangles such a graph can have. The answer is $O(E^{1.5})$ and the maximum occurs with it’s a clique.
Now my question is what happens if the size of maximum clique is fixed, i.e., I cannot have a clique of full size to maximize triangles. Then, under this constraint, how does the optimizer graph (one with maximum triangles) look like?
The claim I have is that the graph should be a union of multiple cliques of appropriate size. But I cannot prove that the maximizer is Union of cliques of appropriate sizes who are independent of each other.