On non-forking spectra

TL;DR

This paper investigates the non-forking spectrum in model theory, classifies its possible values, and develops methods to construct theories with specific spectra, addressing open questions about NIP and linear orders.

Contribution

It provides a classification of non-forking spectra, introduces a construction technique for theories with prescribed spectra, and answers key open questions in the field.

Findings

Possible non-forking spectrum values are limited.

Constructed theories with specific non-forking spectra.

Negatively answered Adler's question on NIP and bounded non-forking.

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.