Infinite Stable Graphs With Large Chromatic Number II

TL;DR

This paper proves a version of Taylor's conjecture for stable graphs with very high chromatic number, showing they contain all finite subgraphs of a certain structure and have extensions with unbounded chromatic number.

Contribution

It generalizes Ehrenfeucht-Mostowski models to an infinitary setting, providing a new characterization of stability and completing previous work on infinite stable graphs.

Findings

Stable graphs with chromatic number > beth_2(aleph_0) contain all finite subgraphs of Sh_n(ω).

Such graphs have elementary extensions with unbounded chromatic number.

Introduces a new model-theoretic construction generalizing Ehrenfeucht-Mostowski models.

Abstract

We prove a version of the strong Taylor's conjecture for stable graphs: if $G$ is a stable graph whose chromatic number is strictly greater than $\beth_2(\aleph_0)$ then $G$ contains all finite subgraphs of Sh$_n(\omega)$ and thus has elementary extensions of unbounded chromatic number. This completes the picture from our previous work. The main new model theoretic ingredient is a generalization of the classical construction of Ehrenfeucht-Mostowski models to an infinitary setting, giving a new characterization of stability.