# Standard monomials and extremal point sets

## TL;DR

This paper extends the concept of shattering-extremality from set systems to general finite point sets, providing algebraic characterizations and strengthening existing results in the field.

## Contribution

It generalizes the algebraic characterization of shattering-extremality from set systems to finite point sets, broadening the theoretical framework.

## Key findings

- Extended shattering-extremality to finite point sets
- Provided algebraic criteria for extremality in the new setting
- Strengthened previous results by Li, Zhang, and Dong

## Abstract

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 $|\text{Sh}(\mathcal{F})|=|\mathcal{F}|$. M\'esz\'aros and R\'onyai, 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.