## Abstract

Non-forking is one of the most important notions in modern model theory capturing the idea of a generic extension of a type (which is a far-reaching generalization of the concept of a generic point of a variety). To a countable first-order theory we associate its non-forking spectrum-a function of two cardinals κ and λ giving the supremum of the possible number of types over a model of size λ that do not fork over a submodel of size κ. This is a natural generalization of the stability function of a theory. We make progress towards classifying the non-forking spectra. On the one hand, we show that the possible values a non-forking spectrum may take are quite limited. On the other hand, we develop a general technique for constructing theories with a prescribed non-forking spectrum, thus giving a number of examples. In particular, we answer negatively a question of Adler whether NIP is equivalent to bounded non-forking. In addition, we answer a question of Keisler regarding the number of cuts a linear order may have. Namely, we show that it is possible that ded κ < (ded κ)^{ω}.

Original language | American English |
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Pages (from-to) | 2821-2848 |

Number of pages | 28 |

Journal | Journal of the European Mathematical Society |

Volume | 18 |

Issue number | 12 |

DOIs | |

State | Published - 2016 |

### Bibliographical note

Publisher Copyright:© European Mathematical Society 2016.

## Keywords

- Cardinal arithmetic
- Circularization
- Dedekind cuts
- Dividing
- Forking
- NIP
- NTP2