Given a fixed graph $H$ that embeds in a surface $\Sigma$, what is the maximum number of copies of $H$ in an $n$-vertex graph $G$ that embeds in $\Sigma$? We show that the answer is $\Theta(n^{f(H)})$, where $f(H)$ is a graph invariant called the `flap-number' of $H$, which is independent of $\Sigma$. This simultaneously answers two open problems posed by Eppstein (1993). When $H$ is a complete graph we give more precise answers.

翻译：鉴于一个固定的硬面美元($H美元)嵌入面面$\西格玛美元,那么以美元嵌入面面面$($G美元)的硬面图中,以美元嵌入的硬面图中,H$($G美元)的最大副本数是多少?我们显示答案是$\Theta(n ⁇ f(H))$($f(H)美元),其中f(H)美元是一个称为$($)的“滚动数字”的硬体图,它独立于$(Sigma)美元。这同时解决了Eppstein(1993年)提出的两个尚未解决的问题。当H美元是一个完整的图表时,我们给出了更准确的答案。