Naming an indiscernible sequence in NIP theories

In this short note we show that if we add predicate for a dense complete indiscernible sequence in a dependent theory then the result is still dependent. This answers a question of Baldwin and Benedikt and implies that every unstable dependent theory has a dependent expansion interpreting linear order.Comment: 3 page

SETS DEFINABLE IN ORDERED ABELIAN GROUPS OF FINITE BURDEN (Model theoretic aspects of the notion of independence and dimension)

In this note, we survey some recent results on definable sets in ordered Abelian groups of finite burden, focusing on topological and arithmetical tameness properties. In the burden 2 case, and assuming definably completeness, definable discrete subsets of the universe can be characterized as those which are definable in an expansion which is elementarily equivalent to (ℝ;<, +, ℤ). We end with some open questions and possible directions for future research

Externally definable sets and dependent pairs

We prove that externally definable sets in first order NIP theories have honest definitions, giving a new proof of Shelah's expansion theorem. Also we discuss a weak notion of stable embeddedness true in this context. Those results are then used to prove a general theorem on dependent pairs, which in particular answers a question of Baldwin and Benedikt on naming an indiscernible sequence.Comment: 17 pages, some typos and mistakes corrected, overall presentation improved, more details for the examples are give

On uniform definability of types over finite sets

In this paper, using definability of types over indiscernible sequences as a template, we study a property of formulas and theories called "uniform definability of types over finite sets" (UDTFS). We explore UDTFS and show how it relates to well-known properties in model theory. We recall that stable theories and weakly o-minimal theories have UDTFS and UDTFS implies dependence. We then show that all dp-minimal theories have UDTFS.Comment: 17 pages, 0 figure

Dp-minimality: basic facts and examples

We study the notion of dp-minimality, beginning by providing several essential facts, establishing several equivalent definitions, and comparing dp-minimality to other minimality notions. 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.Comment: 19 pages; simplified proof for the p-adic