Colouring lines in projective space

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.