Infinite Cliques in Simple and Stable Graphs

TL;DR

This paper investigates when a graph $G$ with $|G|=\mu^+$ and $\chi(G)\ge\mu^+$ must contain large complete subgraphs, under tameness assumptions. It develops two main results: (i) if the theory of $\langle G,E\rangle$ is simple, then $G$ has cliques of every finite size; (ii) if the edge relation $E$ is stable, then $G$ contains an infinite clique of size $\mu^+$. The authors also unify these findings through the chromatic-spectrum function $\mathrm{ch}_T(\mu)$ and analyze the Hajnal–Komjáth example, which has IP and is not simple, illustrating the necessity of tameness. Overall, the work clarifies how stability and simplicity influence the interplay between chromatic number and clique structure in large graphs, with implications for Taylor-type conjectures in tame theories.

Abstract

Suppose that $G$ is a graph of cardinality $μ^+$ with chromatic number $χ(G)\geq μ^+$. One possible reason that this could happen is if $G$ contains a clique of size $μ^+$. 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.

Theorems & Definitions (46)

• Remark 1.2
• proof
• Definition 2.2
• proof
• Example 2.4
• Example 2.5: Symmetric Shift Graph
• Remark 2.7
• Lemma 2.8
• proof
• Lemma 2.9