# Sh:1007

• Chernikov, A., Kaplan, I., & Shelah, S. (2016). On non-forking spectra. J. Eur. Math. Soc. (JEMS), 18(12), 2821–2848.
• 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 \kappa and \lambda giving the supremum of the possible number of types over a model of size \lambda that do not fork over a sub-model of size \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}\kappa<\left({ded}\kappa\right)^{ \omega}.

• published version (28p)
Bib entry
@article{Sh:1007,
author = {Chernikov, Artem and Kaplan, Itay and Shelah, Saharon},
title = {{On non-forking spectra}},
journal = {J. Eur. Math. Soc. (JEMS)},
fjournal = {Journal of the European Mathematical Society (JEMS)},
volume = {18},
number = {12},
year = {2016},
pages = {2821--2848},
issn = {1435-9855},
mrnumber = {3574578},
mrclass = {03C45 (03C55 03C95)},
doi = {10.4171/JEMS/654},
note = {\href{https://arxiv.org/abs/1205.3101}{arXiv: 1205.3101}},
arxiv_number = {1205.3101}
}