Almost covers of finite sets of points

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.