For every finite family $\mathcal{F}$ of graphs, is there a single $G \in \mathcal{F}$ with $\mathrm{ex}(n;G) \ll_{\mathcal{F}} \mathrm{ex}(n;\mathcal{F})$? A counterexample refutes the Erdős-Simonovits compactness conjecture.

If $\mathcal{F}$ is a finite set of finite graphs then $\mathrm{ex}(n;\mathcal{F})$ is the maximum number of edges a graph on $n$ vertices can have without containing any subgraphs from $\mathcal{F}$. Note that it is trivial that $\mathrm{ex}(n;\mathcal{F})\leq \mathrm{ex}(n;G)$ for every $G\in\mathcal{F}$. Is it true that, for every $\mathcal{F}$, there exists $G\in\mathcal{F}$ such that $$\mathrm{ex}(n;G)\ll_{\mathcal{F}}\mathrm{ex}(n;\mathcal{F})?$$

This is the Erdős–Simonovits compactness conjecture. The answer is no: OpenAI [OpenAI26] give a family of connected bipartite graphs, none of them acyclic, for which no single member controls the family extremal number. See erdos_180.variants.counterexample.