We prove that (Z, +, 0) has no proper superstable expansions of finite Lascar rank. Nevertheless, this structure equipped with a predicate defining powers of a given natural number is superstable of Lascar rank ω. Additionally, our methods yield other superstable expansions such as (Z, +, 0) equipped with the set of factorial elements.2010 Mathematics Subject Classification. 03C45.

In this note we develop and clarify some of the basic combinatorial properties of the new notion of n-dependence (for 1 ≤ n < ω) recently introduced by Shelah [She07]. In the same way as dependence of a theory means its inability to encode a bipartite random graph with a definable edge relation, n-dependence corresponds to the inability to encode a random (n + 1)-partite (n + 1)-hypergraph with a definable edge relation. Most importantly, we characterize n-dependence by counting ϕ-types over finite sets (generalizing Sauer-Shelah lemma and answering a question of Shelah from [She05]) and in terms of the collapse of random ordered (n + 1)-hypergraph indiscernibles down to order-indiscernibles (which implies that the failure of n-dependence is always witnessed by a formula in a single free variable).

Abstract. We give an example of a finite rank, in fact ℵ 1 -categorical, theory where the canonical base property (CBP ) fails. In fact we give a "group-like" example in a sense that we will describe below. We also prove, in a finite Morley rank context, that if all definable Galois groups are "rigid" then T has the CBP .

In [3], Hrushovski and the authors proved, in a certain finite rank environment, that rigidity of definable Galois groups implies that T has the canonical base property in a strong form; " internality to" being replaced by "algebraicity in". In the current paper we give a reasonably robust definition of the "strong canonical base property" in a rather more general finite rank context than [3], and prove its equivalence with rigidity of the relevant definable Galois groups. The new direction is an elaboration on the old result that 1-based groups are rigid.

Non-n-ampleness as defined by Pillay and Evans is preserved under analysability. Generalizing this to a more general notion of Sigma-ampleness, we obtain an immediate proof for all simple theories of CHatzidakis weak Canonical Base Property (CBP) for types of finite SU-rank. This is then applied to the special case of groups