A Hilton-Milner Theorem for Vector Spaces

Abstract

We show for $k \geq 2$ that if $q\geq 3$ and $n \geq 2k+1$, or $q=2$ and $n \geq 2k+2$, then any intersecting family ${\cal F}$ of $k$-subspaces of an $n$-dimensional vector space over $GF(q)$ with $\bigcap_{F \in {\cal F}} F=0$ has size at most $\left[{n-1\atop k-1}\right]-q^{k(k-1)}\left[{n-k-1\atop k-1}\right]+q^k$. This bound is sharp as is shown by Hilton-Milner type families. As an application of this result, we determine the chromatic number of the corresponding $q$-Kneser graphs.