On base sizes for almost simple primitive groups

Let $G \leqslant {\rm Sym}(\Omega)$ be a finite almost simple primitive permutation group, with socle $G_0$ and point stabilizer $H$. A subset of $\Omega$ is a base for $G$ if its pointwise stabilizer is trivial; the base size of $G$, denoted $b(G)$, is the minimal size of a base. We say that $G$ is standard if $G_0 = A_n$ and $\Omega$ is an orbit of subsets or partitions of $\{1, \ldots, n\}$, or if $G_0$ is a classical group and $\Omega$ is an orbit of subspaces (or pairs of subspaces) of the natural module for $G_0$. The base size of a standard group can be arbitrarily large, in general, whereas the situation for non-standard groups is rather more restricted. Indeed, we have $b(G) \leqslant 7$ for every non-standard group $G$, with equality if and only if $G$ is the Mathieu group ${\rm M}_{24}$ in its natural action on $24$ points. In this paper, we extend this result by classifying the non-standard groups with $b(G)=6$. The main tools include recent work on bases for actions of simple algebraic groups, together with probabilistic methods and improved fixed point ratio estimates for exceptional groups of Lie type.
Comment: 27 pages; to appear in J. Algebra