Intersecting families of vector spaces with maximum covering number

Abstract

Let $V$ be an $n$-dimensional vector space over the finite field ${\mathbb{F}}_q$. Suppose that $\mathcal{F}$ is an intersecting family of $m$-dimensional subspaces of $V$. The covering number of $\mathcal{F}$ is the minimum dimension of a subspace of $V$ which intersects all elements of $\mathcal{F}$. In this paper, we give the tight upper bound for the size of $\mathcal{F}$ whose covering number is $m$, and describe the structure of $\mathcal{F}$ which reaches the upper bound. Moreover, we determine the structure of an maximum intersecting family of singular linear space with the maximum covering number.

8 pages