Let $Γ_4$ 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 $α:=(q^2+q+1)(q^3+2q^2+q+1)(q+1)^2$ is the independence number of $Γ_4$ and the geometric structure of independent sets with $α$ vertices is described.