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 $\mathrm{ex}(n;\mathcal{F})\le \mathrm{ex}(n;G)$ for every $G\in \mathcal{F}$. Is it true that, for every $\mathcal{F}$, there exists $G\in \mathcal{F}$ such that$$\mathrm{ex}(n;G){\ll }_{\mathcal{F}}\mathrm{ex}(n;\mathcal{F})?$$

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

A very small counterexample is [ \mathcal F={K_{1,2},,2K_2}, ] where (K_{1,2}) is a “cherry” (two edges sharing a vertex) and (2K_2) is a matching of two disjoint edges.

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.