On the Height and Relational Complexity of a Finite Permutation Group

Let G be a permutation group on a set Ω of size t. We say that Λ⊆Ω is an independent set if its pointwise stabilizer is not equal to the pointwise stabilizer of any proper subset of Λ . We define the height of G to be the maximum size of an independent set, and we denote this quantity H(G) . In this paper, we study H(G) for the case when G is primitive. Our main result asserts that either H(G)<9logt or else G is in a particular well-studied family (the primitive large–base groups). An immediate corollary of this result is a characterization of primitive permutation groups with large relational complexity, the latter quantity being a statistic introduced by Cherlin in his study of the model theory of permutation groups. We also study I(G) , the maximum length of an irredundant base of G, in which case we prove that if G is primitive, then either I(G)<7logt or else, again, G is in a particular family (which includes the primitive large–base groups as well as some others).