On base sizes for symmetric groups

A base of a permutation group G on a set Ω is a subset B of Ω such that the pointwise stabilizer of B in G is trivial. The base size of G, denoted by b(G), is the minimal cardinality of a base. Let G = S_n or A_n acting primitively on a set with point stabilizer H. In this note, we prove that if H acts primitively on {1, . . . , n}, and does not contain A_n, then b(G) = 2 for all n > 12. Combined with a theorem of James, this completes the classification of primitive actions of alternating and symmetric groups which admit a base of size 2.