On base sizes for actions of finite classical groups

Let G be a finite almost simple classical group and let Ω be a faithful primitive non-standard G-set. A subset of Ω is a base for G if its pointwise stabilizer in G is trivial. Let b(G) be the minimal size of a base for G. A well- known conjecture of Cameron and Kantor asserts that there exists an absolute constant c such that b(G) is at most c for all such groups G, and the existence of such an undetermined constant has been established by Liebeck and Shalev. In this paper we prove that either b(G) is at most 4, or G = U6(2).2, G_{a} = U4(3).2^2 and b(G) = 5. The proof is probabilistic, using bounds on fixed point ratios.