Good Frames in the Hart-Shelah Example

TL;DR

This paper studies the Hart-Shelah example, an AEC with specific categoricity properties, revealing the precise level of type-shortness and constructing a novel good frame that lacks the uniqueness triples property.

Contribution

It provides the first example of a good frame that fails the existence property for uniqueness triples within the Hart-Shelah AEC.

Findings

Identified the exact amount of type-shortness in the example

Constructed a good ffah_{n-1} frame that fails the uniqueness triples property

Developed new tools for building and analyzing good frames

Abstract

For a fixed natural number $n \geq 1$, the Hart-Shelah example is an abstract elementary class (AEC) with amalgamation that is categorical exactly in the infinite cardinals less than or equal to $\aleph_n$. We investigate recently-isolated properties of AECs in the setting of this example. We isolate the exact amount of type-shortness holding in the example and show that it has a type-full good $\aleph_{n-1}$-frame which fails the existence property for uniqueness triples. This gives the first example of such a frame. Along the way, we develop new tools to build and analyze good frames.