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We consider the structure of $H$-free subgraphs of graphs with high minimaldegree. We prove that for every $k>m$ there exists an$\epsilon:=\epsilon(k,m)>0$ so that the following holds. For every graph $H$with chromatic number $k$ from which one can delete an edge and reduce thechromatic number, and for every graph $G$ on $n>n_0(H)$ vertices in which alldegrees are at least $(1-\epsilon)n$, any subgraph of $G$ which is $H$-free andcontains 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 ofcopies of balanced blowups of complete graphs.

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