  Let $\Gamma$ be the graph whose vertices are the chambers of the finite
projective space $PG(3,q)$ with two vertices being adjacent when the
corresponding chambers are in general position. It is known that the
independence number of this graph is $(q^2+q+1)(q+1)^2$. For $q\ge 43$ we
determine the largest independent set of $\Gamma$ and show that every maximal
independent set that is not a largest one has at most constant times $q^3$
elements. For $q\ge 47$, this information is then used to show that $\Gamma$
has chromatic number $q^2+q$. Furthermore, for many families of generalized
quadrangles we prove similar results for the graph that is built in the same
way on the chambers of the generalized quadrangle.
