Related papers: Existential characterizations of monadic NIP

We give several characterizations of when a complete first-order theory $T$ is monadically NIP, i.e. when expansions of $T$ by arbitrary unary predicates do not have the independence property. The central characterization is a condition on…

We prove various results around indiscernibles in monadically NIP theories. First, we provide several characterizations of monadic NIP in terms of indiscernibles, mirroring previous characterizations in terms of the behavior of finite…

We prove that for any monotone class of finite relational structures, the first-order theory of the class is NIP in the sense of stability theory if, and only if, the collection of Gaifman graphs of structures in this class is nowhere…

Given a complete theory $T$ and a subset $Y \subseteq X^k$, we precisely determine the {\em worst case complexity}, with respect to further monadic expansions, of an expansion $(M,Y)$ by $Y$ of a model $M$ of $T$ with universe $X$. In…

We continue investigating the structure of externally definable sets in NIP theories and preservation of NIP after expanding by new predicates. Most importantly: types over finite sets are uniformly definable; over a model, a family of…

A theory is NIP (resp. stable) if and only if every formula with parameters in two single variables is NIP (resp. does not have the order property).

We introduce the notion of hereditary G-compactness (with respect to interpretation). We provide a sufficient condition for a poset to not be hereditarily G-compact, which we use to show that any linear order is not hereditarily G-compact.…

A relevant thesis is that for the family of complete first order theories with NIP (i.e. without the independence property) there is a substantial theory, like the family of stable (and the family of simple) first order theories. We examine…

We begin a systematic development of structure theory for a first order theory, which is stable over a monadic predicate. We show that stability over a predicate implies quantifier free definability of types over stable sets, introduce an…

We generalize the Unstable Formula Theorem characterization of stable theories from \citep{sh78}: that a theory $T$ is stable just in case any infinite indiscernible sequence in a model of $T$ is an indiscernible set. We use a generalized…

We study one way in which stable phenomena can exist in an NIP theory. We start by defining a notion of 'pure instability' that we call 'distality' in which no such phenomenon occurs. O-minimal theories and the p-adics for example are…

Fix a weakly minimal (i.e., superstable $U$-rank $1$) structure $\mathcal{M}$. Let $\mathcal{M}^*$ be an expansion by constants for an elementary substructure, and let $A$ be an arbitrary subset of the universe $M$. We show that all…

We show that every unstable NIP theory admits a V-definable linear quasi-order, over a finite set of parameters. In particular, if the theory is omega-categorical, then it interprets an infinite linear order. This partially answers a…

Combining two results from machine learning theory we prove that a formula is NIP if and only if it satisfies uniform definability of types over finite sets (UDTFS). This settles a conjecture of Laskowski.

In this note we study sets of NIP formulas in some theories of fields and valued fields, with a special focus on the sets of quantifier-free and existential formulas. First, we give a new proof of the fact that Separably Closed Valued…

We try to understand complete types over a somewhat saturated model of a complete first order theory which is dependent (previously called NIP), by "decomposition theorems for such types". Our thesis is that the picture of dependent theory…

We prove that any type in an NIP theory can be decomposed into a stable part (a generically stable partial type) and a distal-like quotient.

Let K be an algebraically bounded structure and T be its theory. If T is model complete, then the theory of K endowed with a derivation, denoted by $T^{\delta}$, has a model completion. Additionally, we prove that if the theory T is…

We characterise the existentially closed models of the theory of exponential fields. They do not form an elementary class, but can be studied using positive logic. We find the amalgamation bases and characterise the types over them. We…

We introduce the class of unshreddable theories, which contains the simple and NIP theories, and prove that such theories have exactly saturated models in singular cardinals, satisfying certain set-theoretic hypotheses. We also give…