On uncountable hypersimple unidimensional theories

TL;DR

This paper generalizes a dichotomy in model theory, showing that uncountable hypersimple unidimensional theories are supersimple unless they are essentially 1-based, extending previous results to arbitrary languages and topologies.

Contribution

It extends the dichotomy between 1-basedness and supersimplicity to uncountable hypersimple unidimensional theories in arbitrary languages without topological restrictions.

Findings

Uncountable hypersimple unidimensional theories are supersimple unless essentially 1-based.

The dichotomy holds in arbitrary languages with no topology restrictions.

A stronger version of the dichotomy is proved for countable languages.

Abstract

We extend a dichotomy between 1-basedness and supersimplicity proved in a previous paper. The generalization we get is to arbitrary language, with no restrictions on the topology (we do not demand type-definabilty of the open set in the definition of essential 1-basedness). We conclude that every (possibly uncountable) hypersimple unidimensional theory that is not s-essentially 1-based by means of the forking topology is supersimple. We also obtain a strong version of the above dichotomy in the case where the language is countable.