Extremal subspace covers in finite vector spaces

Abstract

Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb F_q$, where $n\ge 2$. It is classical that $q+1$ proper subspaces are necessary and sufficient to cover $V$. We study extremal refinements of this covering theorem. For $1\le m\le q+1$, we determine the maximum possible size of the union of $m$ proper subspaces of $V$, proving that $$ \max_{W_1,\ldots,W_m<V} \left|W_1\cup\cdots\cup W_m\right| = q^{n-2}\bigl(1+m(q-1)\bigr). $$ We also classify all equality cases: for $m\ge 2$, equality holds precisely when the subspaces are distinct hyperplanes containing a common codimension-two subspace. As consequences, we obtain a structural classification of minimum covers of $V$ by proper subspaces and a sharp defect estimate for unions of $q$ proper subspaces. We then introduce basis-blocking families, namely families of proper subspaces whose union meets every basis of $V$. We prove that the minimum size of such a family is $q$ and classify all extremal families of this size. Finally, we establish the affine analogue and give an elementary recognition criterion for extremal families. Together, these results provide a unified extremal-combinatorial description of coverings of finite vector spaces by proper subspaces.