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Extremal subspace covers in finite vector spaces
Combinatorics · Submitted 5 Oct 2026
abstract compiled · 2026-10-07 · no language-model review
The authors' main claim Together, these results provide a unified extremal-combinatorial description of coverings of finite vector spaces by proper subspaces.
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The abstract, sentence by sentence
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 $V$ [@Jamison1977; @Khare2009; @Clark2012]. 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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb F_q$, where $n\ge 2$. | background 1.00 | 0.06 | 0.01 | asserted 0.67 |
| s1 | It is classical that $q+1$ proper subspaces are necessary and sufficient to cover $V$ [@Jamison1977; @Khare2009; @Clark2012]. | background 1.00 | 0.03 | 0.06 | asserted 1.00 |
| s2 | We study extremal refinements of this covering theorem. | contribution 0.97 | 0.54 | 0.04 | asserted 0.51 |
| s3 | For $1\le m\le q+1$, we determine the maximum possible size of the union of $m$ proper subspaces of $V$, proving that $$ \max_{W_1,\ldots,W_m<V} \left|W_1\cup\cdots\cup W_m\right| = q^{n-2}\bigl(1+m(q-1)\bigr). | result 0.98 | 0.95 | 0.63 | asserted 1.00 |
| s4 | $$ 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. | result 0.99 | 0.93 | 0.24 | asserted 1.00 |
| s5 | 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. | result 0.51 | 0.94 | 0.24 | asserted 1.00 |
| s6 | We then introduce basis-blocking families, namely families of proper subspaces whose union meets every basis of $V$. | contribution 1.00 | 0.92 | 0.05 | asserted 0.89 |
| s7 | We prove that the minimum size of such a family is $q$ and classify all extremal families of this size. | result 0.94 | 0.96 | 0.12 | asserted 1.00 |
| s8 | Finally, we establish the affine analogue and give an elementary recognition criterion for extremal families. | contribution 0.73 | 0.95 | 0.08 | asserted 1.00 |
| s9 | Together, these results provide a unified extremal-combinatorial description of coverings of finite vector spaces by proper subspaces. | interpretation 0.99 | 0.78 | 0.46 | asserted 0.99 |
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Original 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.
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