On the height and relational complexity of a finite permutation group

Let G be a permutation group on a set of size t. We say that is an independent set if its pointwise stabilizer is not equal to the pointwise stabilizer of any proper subset of. We define the height of G to be the maximum size of an independent set, and we denote this quantity. In this paper, we study for the case when G is primitive. Our main result asserts that either {H}(G) 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, the maximum length of an irredundant base of G, in which case we prove that if G is primitive, then either I(G) 7 log t or else, again, G is in a particular family (which includes the primitive large-base groups as well as some others).