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]Fe:FF = 0 has size at most [nkz11] - qk(k~1) nkk11] + 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.