Almost covers of finite sets of points

TL;DR

This paper establishes a lower bound on the size of minimal almost covers of finite point sets in affine space, generalizing existing theorems using Gr"obner basis theory.

Contribution

It provides a new lower bound for almost covers of finite point sets, extending previous results with a proof based on Gr"obner bases.

Findings

Derived a lower bound for minimal almost covers.

Generalized Sziklai and Weiner's Theorem.

Used Gr"obner basis theory for the proof.

Abstract

Let $\mbox{$\cal V$} \subseteq {\mathbb F}^n$ be a finite set of points in an affine space. A finite set of affine hyperplanes $\{H_1, \ldots ,H_m\}$ is said to be an almost cover of $\mbox{$\cal V$}$ and $\mathbf{v}$, if their union $\cup_{j=1}^m H_j$ contains $\mbox{$\cal V$}\setminus \{\mathbf{v}\}$ but does not contain $\mathbf{v}$. We give here a lower bound for the size of a minimal almost cover of $\mbox{$\cal V$}$ and $\mathbf{v}$ in terms of the size of $\mbox{$\cal V$}$ and the dimension $n$. We prove a generalization of Sziklai and Weiner's Theorem. Our simple proof is based on Gr\"obner basis theory.