Shelah's eventual categoricity conjecture in tame AECs with primes

TL;DR

This paper proves a new case of Shelah's eventual categoricity conjecture for tame AECs with primes, extending previous results and applying the method to homogeneous model theory, showing categoricity transfers under weaker assumptions.

Contribution

It establishes Shelah's conjecture for $H_2$-tame AECs with primes over sets, relaxing previous locality assumptions, and applies the proof technique to homogeneous model theory.

Findings

Categoricity in some $ eq H_2$ implies categoricity in all $ eq H_2$ for the class.

The result extends Shelah's conjecture to a broader class of AECs with primes.

The method applies to homogeneous diagrams, confirming categoricity transfer in that context.

Abstract

A new case of Shelah's eventual categoricity conjecture is established: $\mathbf{Theorem}$ Let $K$ be an AEC with amalgamation. Write $H_2 := \beth_{\left(2^{\beth_{\left(2^{\text{LS} (K)}\right)^+}}\right)^+}$. Assume that $K$ is $H_2$-tame and $K_{\ge H_2}$ has primes over sets of the form $M \cup \{a\}$. If $K$ is categorical in some $\lambda > H_2$, then $K$ is categorical in all $\lambda' \ge H_2$. The result had previously been established when the stronger locality assumptions of full tameness and shortness are also required. An application of the method of proof of the theorem is that Shelah's categoricity conjecture holds in the context of homogeneous model theory (this was known, but our proof gives new cases): $\mathbf{Theorem}$ Let $D$ be a homogeneous diagram in a first-order theory $T$. If $D$ is categorical in a $\lambda > |T|$, then $D$ is categorical in all $\lambda' \ge \min (\lambda, \beth_{(2^{|T|})^+})$.