Infinite Stable Graphs With Large Chromatic Number

TL;DR

This paper demonstrates that certain infinite stable graphs with large chromatic numbers contain all finite subgraphs of shift graphs, revealing deep structural properties linked to stability and interpretability in model theory.

Contribution

It establishes that omega-stable and superstable graphs with large chromatic numbers contain all finite shift graph subgraphs, extending known results in model-theoretic graph theory.

Findings

Graphs with large chromatic number contain all finite shift graph subgraphs.

For omega-stable graphs with U-rank ≤ 2, only shift graphs with n ≤ 2 are contained.

Results extend to graphs interpretable in stationary stable theories.

Abstract

We prove that if $G=(V,E)$ is an $\omega$-stable (respectively, superstable) graph with $\chi(G)>\aleph_0$ (respectively, $2^{\aleph_0}$) then $G$ contains all the finite subgraphs of the shift graph $\text{Sh}_n(\omega)$ for some $n$. We prove a variant of this theorem for graphs interpretable in stationary stable theories. Furthermore, if $G$ is $\omega$-stable with $\mathrm{U}(G)\leq 2$ we prove that $n\leq 2$ suffices.