The base size of the symmetric group acting on subsets

Abstract

A base for a permutation group $G$ acting on a set $\Omega$ is a subset $\mathcal{B}$ of $\Omega$ such that the pointwise stabiliser $G_{(\mathcal{B})}$ is trivial. Let $n$ and $r$ be positive integers with $n>2r$. The symmetric and alternating groups $\mathrm{S}_n$ and $\mathrm{A}_n$ admit natural primitive actions on the set of $r$-element subsets of $\{1,2,\dots, n\}$. Building on work of Halasi [6], we provide explicit expressions for the base sizes of all of these actions, and hence determine the base size of all primitive actions of $\mathrm{S}_n$ and $\mathrm{A}_n$.