Dependent theories and the generic pair conjecture

TL;DR

This paper explores the structure of dependent theories in model theory, aiming to understand types over saturated models and addressing the generic pair conjecture for measurable cardinals.

Contribution

It introduces decomposition theorems for types in dependent theories and advances the proof of the generic pair conjecture in the context of measurable cardinals.

Findings

Developed decomposition theorems for dependent theories

Proved the generic pair conjecture for measurable cardinals

Enhanced understanding of types over saturated models in dependent theories

Abstract

We try to understand complete types over a somewhat saturated model of a complete first order theory which is dependent (previously called NIP), by "decomposition theorems for such types". Our thesis is that the picture of dependent theory is the combination of the one for stable theories and the one for the theory of dense linear order or trees (and first we should try to understand the quite saturated case). As a measure of our progress, we give several applications considering some test questions; in particular we try to prove the generic pair conjecture and do it for measurable cardinals.