On the uniform spread of almost simple linear groups

Let G be a finite group, and let k be a nonnegative integer. We say that G has uniform spread k if there exists a fixed conjugacy class C in G with the property that for any k nontrivial elements x(1),...,x(k) in G there exists y is an element of C such that G = <x(i), y > for all i. Further, the exact uniform spread of G, denoted by u(G), is the largest k such that G has the uniform spread k property. By a theorem of Breuer, Guralnick, and Kantor, u(G) > 1 for every finite simple group G. Here we consider the uniform spread of almost simple linear groups. Our main theorem states that if G = <PSLn(q),g > is almost simple, then u(G) > 1 (unless G is isomorphic to S_6), and we determine precisely when u(G) tends to infinity as |G| tends to infinity.