JournalsjemsVol. 18, No. 12pp. 2821–2848

On non-forking spectra

  • Artem Chernikov

    University of California at Los Angeles, USA
  • Itay Kaplan

    The Hebrew University of Jerusalem, Israel
  • Saharon Shelah

    The Hebrew University of Jerusalem, Israel
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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\kappa and λ\lambda giving the supremum of the possible number of types over a model of size λ\lambda that do not fork over a sub-model of size κ\kappa. 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\kappa < (dedκ)ω_\kappa) ^{\omega}.

Cite this article

Artem Chernikov, Itay Kaplan, Saharon Shelah, On non-forking spectra. J. Eur. Math. Soc. 18 (2016), no. 12, pp. 2821–2848

DOI 10.4171/JEMS/654