We say that a set system $\mathcal{F}\subseteq 2^{[n]}$ shatters a set $S\subseteq [n]$ if every possible subset of $S$ appears as the intersection of $S$ with some element of $\mathcal{F}$ and we denote by $\text{Sh}(\mathcal{F})$ the family of sets shattered by $\mathcal{F}$. According to the Sauer-Shelah lemma we know that in general, every set system $\mathcal{F}$ shatters at least $|\mathcal{F}|$ sets and we call a set system shattering-extremal if $|\text{Sh}(\mathcal{F})|=|\mathcal{F}|$. Meszaros and Ronyai, among other things, gave an algebraic characterization of shattering-extremality, which offered the possibility to generalize the notion to general finite point sets. Here we extend the results obtained for set systems to this more general setting, and as an application, strengthen a result of Li, Zhang and Dong.

中文翻译：

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标准单项式和极值点集

我们说一个集合系统 $\mathcal{F}\subseteq 2^{[n]}$ 粉碎了一个集合 $S\subseteq [n]$ 如果 $S$ 的每个可能的子集都出现为 $S$ 与$\mathcal{F}$ 的某个元素，我们用 $\text{Sh}(\mathcal{F})$ 表示被 $\mathcal{F}$ 粉碎的集合族。根据 Sauer-Shelah 引理，我们知道一般来说，每个集合系统 $\mathcal{F}$ 至少粉碎 $|\mathcal{F}|$ 集合，如果 $|\text {Sh}(\mathcal{F})|=|\mathcal{F}|$。除其他外，Meszaros 和 Ronyai 给出了粉碎极值的代数表征，这提供了将概念推广到一般有限点集的可能性。在这里，我们将集合系统获得的结果扩展到这个更一般的设置，并作为一个应用，加强 Li、Zhang 和 Dong 的结果。