No Maximal Models from Looking Down

TL;DR

This paper extends Shelah's results on the number of models in Abstract Elementary Classes (AECs), demonstrating that with limited amalgamation, the existence of arbitrarily large extensions applies to more cardinals.

Contribution

It generalizes Shelah's theorem by relaxing amalgamation assumptions, broadening the class of cardinals where models have arbitrarily large extensions.

Findings

Under limited amalgamation, more cardinals exhibit models with arbitrarily large extensions.

The results apply to a larger class of cardinals than previously established.

The paper advances understanding of model existence in AECs with restricted amalgamation.

Abstract

In [Sh893], Shelah proves that (on a stationary set of cardinals) an AEC has not too many models or every model has extensions of arbitrary cardinality. We show that, if we assume limited amalgamation, then the second condition holds for a larger class of cardinals.