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.