Let V be a vector space of dimension v over a field of order q. The
q-Kneser graph has the k-dimensional subspaces of V as its vertices,
where two subspaces α and β are adjacent if and only if
α∩β is the zero subspace. This paper is motivated by the problem
of determining the chromatic numbers of these graphs. This problem is trivial
when k=1 (and the graphs are complete) or when v<2k (and the graphs are
empty). We establish some basic theory in the general case. Then specializing
to the case k=2, we show that the chromatic number is q2+q when v=4 and
(qv−1−1)/(q−1) when v>4. In both cases we characterise the minimal
colourings.Comment: 19 pages; to appear in J. Combinatorial Theory, Series