A Hilton-Milner Theorem for Vector Spaces

We show for k ≥ 3 that if q ≥ 3, n ≥ 2k + 1 or q = 2, n ≥ 2k + 2, then any intersecting family F of k-subspaces of an n-dimensional vector space over GF (q) with ⋂ F∈F F = 0 has size at most [ n−1 k−1 ] − qk(k−1) [ n−k−1 k−1 ] +qk. 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.