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 and giving the supremum of the possible number of types over a model of size that do not fork over a sub-model of size . 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 < (ded.
Cite this article
Artem Chernikov, Itay Kaplan, Saharon Shelah, On non-forking spectra. J. Eur. Math. Soc. 18 (2016), no. 12, pp. 2821–2848DOI 10.4171/JEMS/654