On the chromatic number of q-Kneser graphs

We show that the q-Kneser graph qK2k:k (the graph on the k-subspaces of a 2k-space over GF(q), where two k-spaces are adjacent when they intersect trivially), has chromatic number qk + qk−1 for k = 3 and for k < q log q − q. We obtain detailed results on maximal cocliques for k = 3.