Classical questions in extremal graph theory concern the asymptotics of $operatorname{ex}(G, mathcal{H})$ where $mathcal{H}$ is a fixed family of graphs and $G=G_n$ is taken from a `standard increasing sequence of host graphs $(G_1, G_2, dots)$, most often $K_n$ or $K_{n,n}$. Inverting the question, we can instead ask how large $e(G)$ can be with respect to $operatorname{ex}(G,mathcal{H})$. We show that the standard sequences indeed maximize $e(G)$ for some choices of $mathcal{H}$, but not for others. Many interesting questions and previous results arise very naturally in this context, which also, unusually, gives rise to sensible extremal questions concerning multigraphs and non-uniform hypergraphs.
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