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 ) ⩽ slant 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) · 2, G ω = U 4 (3) · 2 2 and b ( G ) = 5. The proof is probabilistic, using bounds on fixed point ratios.