# Sh:1007

- Chernikov, A., Kaplan, I., & Shelah, S. (2016).
*On non-forking spectra*. J. Eur. Math. Soc. (JEMS),**18**(12), 2821–2848. arXiv: 1205.3101 DOI: 10.4171/JEMS/654 MR: 3574578 -
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)

@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}, mrreviewer = {Sebastien Vasey}, mrclass = {03C45 (03C55 03C95)}, doi = {10.4171/JEMS/654}, note = {\href{https://arxiv.org/abs/1205.3101}{arXiv: 1205.3101}}, arxiv_number = {1205.3101} }