A generalisation of Cameron's base size conjecture

Abstract

Let G⩽Sym(Ω)$G\leqslant {\rm Sym}(\Omega)$ be a finite transitive permutation group with point stabiliser H$H$ . A base for G$G$ is a subset of Ω$\Omega$ whose pointwise stabiliser is trivial, and the minimal cardinality of a base is called the base size of G$G$ , denoted by b(G,Ω)$b(G, \Omega)$ . Equivalently, b(G,Ω)$b(G, \Omega)$ is the minimal positive integer k$k$ such that G$G$ has a regular orbit on the Cartesian product Ωk$\Omega ^k$ . A well‐known conjecture of Cameron from the 1990s asserts that if G$G$ is an almost simple primitive group and H$H$ is a so‐called non‐standard subgroup, then b(G,Ω)⩽7$b(G, \Omega) \leqslant 7$ , with equality if and only if G$G$ is the Mathieu group M24${\rm M}_{24}$ in its natural action of degree 24. This conjecture was settled in a series of papers by Burness et al.