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 < 2 k (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 q 2 + q when v = 4 and ( q v - 1 - 1 ) / ( q - 1 ) when v > 4 . In both cases we characterise the minimal colourings.