  Let $\Gamma_4$ be the graph whose vertices are the chambers of the finite
projective $4$-space PG(4,q), with two vertices being adjacent if the
corresponding chambers are in general position. For $q\geq 749 $ we show that
$\alpha:=(q^2+q+1)(q^3+2q^2+q+1)(q+1)^2$ is the independence number of
$\Gamma_4$ and the geometric structure of independent sets with $\alpha$
vertices is described.
