On non-forking spectra

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 $\kappa$ < (ded${}_{\kappa }{)}^{\omega }$.