Examples in dependent theories

TL;DR

This paper explores properties of dependent theories, providing a counterexample to Shelah's conjecture, analyzing generic pairs, introducing directionality, and examining non-splintering, with applications to dense types in real closed fields.

Contribution

It presents a counterexample to Shelah's conjecture, introduces the concept of directionality, and investigates non-splintering in dependent theories, expanding understanding of their structural properties.

Findings

Counterexample to Shelah's conjecture on indiscernible sequences.

Examples of non-dependent generic pairs.

Different possibilities of directionality in theories.

Abstract

In the first part we show a counterexample to a conjecture by Shelah regarding the existence of indiscernible sequences in dependent theories (up to the first inaccessible cardinal). In the second part we discuss generic pairs, and give an example where the pair is not dependent. Then we define the notion of directionality which deals with counting the number of coheirs of a type and we give examples of the different possibilities. Then we discuss non-splintering, an interesting notion that appears in the work of Rami Grossberg, Andr\'es Villaveces and Monica VanDieren, and we show that it is not trivial (in the sense that it can be different than splitting) whenever the directionality of the theory is not small. In the appendix we study dense types in RCF.