On base sizes for actions of finite classical groups

Let G be a finite almost simple classical group and let Omega be a faithful primitive non-standard G-set. A subset of Omega 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 wellknown conjecture of Cameron and Kantor asserts that there exists an absolute constant c such that b(G) <= 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) <= 4, or G = U-6 (2) center dot 2, G(w) = U-4(3) center dot 2(2) and b(G) = 5. The proof is probabilistic, using bounds on fixed point ratios.