Maximal cocliques and the chromatic number of the Kneser graph on chambers of PG ( 3 , q ) $(3,q)$

Let $\Gamma$ be the graph whose vertices are the chambers of the finite projective 3-space $\text{PG}\left(3,q\right)$, with two vertices being adjacent if and only if the corresponding chambers are in general position. We show that a maximal independent set of vertices of $\Gamma$ contains $q^4+3q^3+4q^2+3q+1$, or $3q^3+5q^2+3q+1$, or at most $3q^3+4q^2+3q+2$ elements. For $q\ge 4$ the structure of the largest maximal independent sets is described. For $q\ge 7$ the structure of the maximal independent sets of the three largest cardinalities is described. Using the cardinality of the second largest maximal independent sets, we show that the chromatic number of $\Gamma$ is $q^2+q$.