A Hilton-Milner theorem for vector spaces

Abstract

We show for k � 2 that if q � 3 and n � 2k + 1, or q = 2 and 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