The base size of the symmetric group acting on subsets

A base for a permutation group G acting on a set Ω is a subset ℬ of Ω such that the pointwise stabiliser G (ℬ) is trivial. Let n and r be positive integers with n>2r. The symmetric and alternating groups S n and A n admit natural primitive actions on the set of r-element subsets of 1, 2, ⋯, n. Building on work of Halasi 8, we provide explicit expressions for the base sizes of all of these actions, and hence determine the base size of all primitive actions of § n and A n.