Infinite Cliques in Simple and Stable Graphs

TL;DR

The paper investigates the existence of large cliques in graphs with high chromatic number, proving their presence in stable graphs and establishing bounds in simple but unstable graphs.

Contribution

It demonstrates that graphs with stable edge relations and high chromatic number necessarily contain large cliques, extending understanding of graph structure in model theory.

Findings

Stable graphs with high chromatic number contain large cliques.

Unstable but simple graphs have finite but unbounded large cliques.

Random graphs do not necessarily contain large cliques despite high chromatic number.

Abstract

Suppose that $G$ is a graph of cardinality $\mu^+$ with chromatic number $\chi(G)\geq \mu^+$. One possible reason that this could happen is if $G$ contains a clique of size $\mu^+$. We prove that this is indeed the case when the edge relation is stable. When $G$ is a random graph (which is simple but not stable), this is not true. But still if in general the complete theory of $G$ is simple, $G$ must contain finite cliques of unbounded sizes.