发表机构
Clayton State University（克莱顿州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究有限向量空间中真子空间覆盖的极值问题，确定了 $m$ 个真子空间并集的最大大小并分类所有等式情形，同时引入基阻塞族并证明其最小大小为 $q$，给出统一极值组合描述。
AI 中文摘要
设 $V$ 是有限域 $\mathbb F_q$ 上的 $n$ 维向量空间，其中 $n\ge 2$。经典结果表明，$q+1$ 个真子空间是覆盖 $V$ 的充分必要条件 [@Jamison1977; @Khare2009; @Clark2012]。我们研究该覆盖定理的极值细化。对于 $1\le m\le q+1$，我们确定了 $V$ 的 $m$ 个真子空间并集的最大可能大小，证明 $$ \max_{W_1,\ldots,W_m<V} \left|W_1\cup\cdots\cup W_m\right| = q^{n-2}\bigl(1+m(q-1)\bigr). $$ 我们还对所有等式情形进行了分类：对于 $m\ge 2$，当且仅当这些子空间是包含一个共同余维二子空间的不同的超平面时等式成立。作为推论，我们获得了 $V$ 由真子空间构成的最小覆盖的结构分类，以及 $q$ 个真子空间并集的尖锐缺陷估计。随后我们引入了基阻塞族，即其并集与 $V$ 的每个基都相交的真子空间族。我们证明了此类族的最小大小为 $q$，并对所有该大小的极值族进行了分类。最后，我们建立了仿射类比，并给出了极值族的一个初等识别准则。这些结果共同为有限向量空间由真子空间覆盖提供了统一的极值组合描述。
英文摘要
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.
CommentsMinor typos have been fixed. 11 pages. To appear in Discrete Mathematics, Algorithms and Applications
