Abstract. The open core of an expansion of a dense linear order is its reduct, in the sense of definability, generated by the collection of all of its open definable sets.

We consider strong expansions of the theory of ordered Abelian groups. We show that the assumption of strength has a multitude of desirable consequences for the structure of definable sets in such theories, in particular as relates to definable infinite discrete sets. We also provide a range of examples of strong expansions of ordered Abelian groups which demonstrate the great variety of such theories.

We study the notion of dp-minimality, beginning by providing several essential facts about dp-minimality, establishing several equivalent definitions for dp-minimality, and comparing dp-minimality to other minimality notions. The majority of the rest of the paper is dedicated to examples. We establish via a simple proof that any weakly o-minimal theory is dp-minimal and then give an example of a weakly o-minimal group not obtained by adding traces of externally definable sets. Next we give an example of a divisible ordered Abelian group which is dp-minimal and not weakly o-minimal. Finally we establish that the field of p-adic numbers is dp-minimal.2 ALFRED DOLICH, JOHN GOODRICK, AND DAVID LIPPEL these facts are inherent in [14] but we isolate them here and provide straightforward proofs. Section 3 is a brief discussion of the relationship of dp-minimality to some other minimality notions. In Section 4 we focus weak on o-minimality-for which see [10]-and show that a weakly o-minimal theory is dp-minimal as well providing an example of a weakly o-minimal group which is not obtained by expanding on o-minimal structure by convex sets. Work in Section 3 as well as results from [9] indicate that a dp-minimal theory expanding that of divisible ordered Abelian groups has some similarity to a weakly o-minimal theory and we may naturally ask whether any such theory is weakly o-minimal. Section 5 provides a negative answer via an example arising form the valued field context. Our final section is dedicated to showing that the theory of the p-adic field is dp-minimal. basic facts on dp-minimalityWe develop several basics facts about dp-minimality. The vast majority of the material found below is inherent in Shelah's paper [14], but typically in the more general context of strong dependence. We provide proofs of these various facts for clarity and ease of exposition. Recall:Definition 2.1. Fix a structure M, An ICT pattern in M consists of a pair of formulae φ(x, y) and ψ(x, y); and sequences {a i : i ∈ ω} and {b i : i ∈ ω} from M so that for all i, j ∈ ω the following is consistent:Remark. Definition 2.1 should more formally be referred to as an ICT pattern of depth two but in this paper we only consider such ICT patterns and thus we omit this extra terminology.Definition 2.2. A theory T is said to be dp-minimal if in no model M |= T is there an ICT pattern.It is often very convenient to use the following definition and fact.Definition 2.3. We say two sequences {a i : i ∈ I} and {b j : j ∈ J} are mutually indiscernible if {a i : i ∈ I} is indiscernible over j∈J b j and {b j : j ∈ J} is indiscernible over i∈I a i . We call an ICT pattern mutually indiscernible if the witnessing sequences are mutually indiscernible. Fact 2.4. T is dp-minimal if and only if in no model M |= T is there a mutually indiscernible ICT pattern.Proof. This is a simple application of compactness and Ramsey's theorem.Before continuing we should mention another alternative characterization of dpminimality. To this end we have the following definition.

Let M be an o-minimal expansion of a densely ordered group and H be a pairwise disjoint collection of dense subsets of M such that H is definably independent in M. We study the structure (M, (H) H∈H ). Positive results include that every open set definable in (M, (H) H∈H ) is definable in M, the structure induced in (M, (H) H∈H ) on any H 0 ∈ H is as simple as possible (in a sense that is made precise), and the theory of (M, (H) H∈H ) eliminates imaginaries and is strongly dependent and axiomatized over the theory of M in the most obvious way. Negative results include that (M, (H) H∈H ) does not have definable Skolem functions and is neither atomic nor satisfies the exchange property. We also characterize (model-theoretic) algebraic closure and thorn forking in such structures. Throughout, we compare and contrast our results with the theory of dense pairs of o-minimal structures.