  Let $\Gamma$ be the graph whose vertices are the chambers of the finite
projective $3$-space PG$(3,q)$, 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 $q\geq 4$ the
structure of the largest maximal independent sets is described. For $q\geq 7$
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 $q^2+q$.
