Almost covers of finite sets of points

Let V⊆ Fⁿ be a finite set of points in an affine space. A finite set of affine hyperplanes ₁, … ,Hₘ\ is said to be an almost cover of V and v, if their union ∪ⱼ₌₁ᵐ Hⱼ contains V\ but does not contain v. We give here a lower bound for the size of a minimal almost cover of V and v in terms of the size of 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.