Sets of elements that pairwise generate a linear group

Let G be any of the groups(P)GL(n,q),(P)SL(n,q). Define a (simple) graph Γ =Γ (G) on the set of elements of G by connecting two vertices by an edge if and only if they generate G. Suppose that n is at least 12. Then the maximum size of a complete subgraph in Γ is equal to the chromatic number of Γ if n ≢2 (mod 4), or if n≡2 (mod 4), q is odd and G =(P)SL(n,q). This work was motivated by a question of Blackburn.