Base sizes of primitive permutation groups

Let G be a permutation group, acting on a set Ω of size n . A subset B of Ω is a base for G if the pointwise stabilizer G ( B ) is trivial. Let b ( G ) be the minimal size of a base for G . A subgroup G of Sym ( n ) is large base if there exist integers m and r ≥ 1 such that Alt ( m ) r ⊴ G ≤ Sym ( m ) ≀ Sym ( r ) , where the action of Sym ( m ) is on k -element subsets of { 1 , ⋯ , m } and the wreath product acts with product action. In this paper we prove that if G is primitive and not large base, then either G is the Mathieu group M 24 in its natural action on 24 points, or b ( G ) ≤ ⌈ log n ⌉ + 1 . Furthermore, we show that there are infinitely many primitive groups G that are not large base for which b ( G ) > log n + 1 , so our bound is optimal.