Almost covers of finite sets of points

Let \mbox{\cal V} \subseteq {\mathbb F}^n be a finite set of points in an affine space. A finite set of affine hyperplanes $\{H_1,\dots ,H_m\}$ 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 Gröbner basis theory.