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