Externally definable sets and dependent pairs II

@article{Chernikov2012ExternallyDS,
  title={Externally definable sets and dependent pairs II},
  author={Artem Chernikov and Pierre Simon},
  journal={Transactions of the American Mathematical Society},
  year={2012},
  volume={367},
  pages={5217-5235}
}
Author(s): Chernikov, A; Simon, P | Abstract: 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 non-forking instances of a formula (with parameters ranging over a type-definable set) can be covered with finitely many invariant types; we give some criteria for the boundedness of an expansion by a new predicate in… 

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References

SHOWING 1-10 OF 48 REFERENCES

Externally definable sets and dependent pairs II

Author(s): Chernikov, A; Simon, P | Abstract: We continue investigating the structure of externally definable sets in NIP theories and preservation of NIP after expanding by new predicates. Most

On uniform definability of types over finite sets

This paper explores UDTFS and shows how it relates to well-known properties in model theory and recalls that stable theories and weakly o-minimal theories have UDTFF and implies dependence.

On NIP and invariant measures

We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [13]. Among key results are

Dependent first order theories, continued

A dependent theory is a (first order complete theory) T which does not have the independence property. A major result here is: if we expand a model of T by the traces on it of sets definable in a

Stable embeddedness and NIP

  • A. Pillay
  • Mathematics
    The Journal of Symbolic Logic
  • 2011
This paper gives some sufficient conditions for a predicate P in a complete theory T to be “stably embedded” and makes use of the theory of strict nonforking and weight in NIP theories.

Strongly dependent theories

We further investigate the class of models of a strongly dependent (first order complete) theory T, continuing [Sh:715], [Sh:783] and related works. Those are properties (= classes) somewhat parallel

Dense pairs of o-minimal structures

The structure of definable sets and maps in dense elementary pairs of o-minimal expansions of ordered abelian groups is described. It turns out that a certain notion of “small definable set” plays a

Stable theories with a new predicate

Every algebraic set intersects U in a finite union of translates of subgroups definable in the group structure of U alone and whence Uind is nothing more than a (divisible) abelian group, which isω–stable.

Definable types in -minimal theories

The goal is to provide a characterization of definable types over r-minimal structures that generalizes van den Dries' results and shows that every type over an M- minimal expansion -of R is definable.

Compression Schemes, Stable Definable Families, and o-Minimal Structures

The combinatorial complexity of any definable family in a structure with a o-minimal theory is bounded by the number of parameters in the defining formula, and extended compression schemes for uniformly definable families corresponding to stable formulas are shown to exist.