On the largest independent sets in the Kneser graph on chambers of PG(4,q)

Let ${\mathrm{\Gamma }}_4$ 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 $q\ge 749$ we show that $\alpha :=(q^2+q+1)(q^3+2q^2+q+1)(q+1{)}^2$ is the independence number of ${\mathrm{\Gamma }}_4$ and the geometric structure of independent sets with $\alpha$ vertices is described.