Tameness and frames revisited

TL;DR

This paper investigates extending good frames in tame abstract elementary classes to longer types, demonstrating that tameness and good frames ensure well-behaved dimension theory and improving frame extension results.

Contribution

It shows that good frames can be extended to types of independent sequences in tame AECs, and links tameness and good frames to Shelah's dimension theory.

Findings

Good frames extend to types of independent sequences.

Tameness and good frames imply well-behaved Shelah dimension.

Improved results on extending frames to larger models.

Abstract

We study the problem of extending an abstract independence notion for types of singletons (what Shelah calls a good frame) to longer types. Working in the framework of tame abstract elementary classes, we show that good frames can always be extended to types of independent sequences. As an application, we show that tameness and a good frame imply Shelah's notion of dimension is well-behaved, complementing previous work of Jarden and Sitton. We also improve a result of the first author on extending a frame to larger models.