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On chromatic number of countable graphs

Hirotaka Kikyo, Koitaro Nakaura, Akito Tsuboi

Abstract

This paper investigates when countable graphs have a finite or an infinite chromatic number through model theoretic methods. For Fraïssé limits, we show that instability forces the chromatic number to be infinite, yielding a complete classification of homogeneous graphs with a finite chromatic number. In contrast, Hrushovski construction always produces graphs of finite chromatic number, though the value can be made arbitrarily large. In tame settings -- such as stable graphs of $U$-rank one and graphs definable in o-minimal structures -- an infinite chromatic number necessarily yields arbitrarily large cliques. These results provide a unified framework connecting structural model theoretic properties with chromatic behavior.

On chromatic number of countable graphs

Abstract

This paper investigates when countable graphs have a finite or an infinite chromatic number through model theoretic methods. For Fraïssé limits, we show that instability forces the chromatic number to be infinite, yielding a complete classification of homogeneous graphs with a finite chromatic number. In contrast, Hrushovski construction always produces graphs of finite chromatic number, though the value can be made arbitrarily large. In tame settings -- such as stable graphs of -rank one and graphs definable in o-minimal structures -- an infinite chromatic number necessarily yields arbitrarily large cliques. These results provide a unified framework connecting structural model theoretic properties with chromatic behavior.
Paper Structure (9 sections, 16 theorems, 15 equations)

This paper contains 9 sections, 16 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Basic Results
  3. Fraïssé Limits
  4. Infinite chromatic numbers
  5. Finite Chromatic Numbers
  6. Hrushovski Construction
  7. Graphs without complex structures
  8. $U$-rank one graphs
  9. o-minimal graphs

Key Result

Theorem A

Let $G$ be a countable homogeneous graph. If $G$ is unstable, then $\chi(G)$ is infinite.

Theorems & Definitions (44)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Definition 1
  • Remark 2
  • Definition 3: Mycielskian
  • proof
  • Proposition 5
  • proof
  • ...and 34 more