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.
A family F ⊆ 2 [n] saturates the monotone decreasing property P if F satisfies P and one cannot add any set to F such that property P is still satisfied by the resulting family. We address the problem of finding the minimum size of a family saturating the k-Sperner property and the minimum size of a family that saturates the Sperner property and that consists only of l-sets and (l + 1)-sets.
We prove a vector space analog of a version of the KruskalKatona theorem due to Lovász. We apply this result to extend Frankl's theorem on r-wise intersecting families to vector spaces. In particular, we obtain a short new proof of the Erdős-Ko-Rado theorem for vector spaces.
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