H-free subgraphs of dense graphs maximizing the number of cliques and their blow-ups

We consider the structure of H-free subgraphs of graphs with high minimal degree. We prove that for every k>m there exists an ϵ≔ϵ(k,m)>0 so that the following holds. For every graph H with chromatic number k from which one can delete an edge and reduce the chromatic number, and for every graph G on n>n₀(H) vertices in which all degrees are at least (1−ϵ)n, any subgraph of G which is H-free and contains the maximum number of copies of the complete graph K_m is (k−1)-colorable. We also consider several extensions for the case of a general forbidden graph H of a given chromatic number, and for subgraphs maximizing the number of copies of balanced blowups of complete graphs.