# The Ramsey theory of the universal homogeneous triangle-free graph

## TL;DR

This paper establishes the Ramsey theory for the universal homogeneous triangle-free graph, showing that for any finite triangle-free graph, there is a bound on the number of colors needed in a homogeneous substructure, using new broadscale techniques.

## Contribution

It provides the first comprehensive Ramsey-theoretic result for a homogeneous structure omitting a non-trivial finite subgraph, introducing new coding and tree construction methods.

## Key findings

- Proves existence of bounded colorings in homogeneous subgraphs
- Develops new broadscale techniques for coding $ ext{H}_3$
- Extends Ramsey theory to triangle-free homogeneous graphs

## Abstract

The universal homogeneous triangle-free graph, constructed by Henson and denoted $\mathcal{H}_3$, is the triangle-free analogue of the Rado graph. While the Ramsey theory of the Rado graph has been completely established, beginning with Erd\H{o}s-Hajnal-Pos\'{a} and culminating in work of Sauer and Laflamme-Sauer-Vuksanovic, the Ramsey theory of $\mathcal{H}_3$ had only progressed to bounds for vertex colorings (Komj\'{a}th-R\"{o}dl) and edge colorings (Sauer). This was due to a lack of broadscale techniques. We solve this problem in general: For each finite triangle-free graph $G$, there is a finite number $T(G)$ such that for any coloring of all copies of $G$ in $\mathcal{H}_3$ into finitely many colors, there is a subgraph of $\mathcal{H}_3$ which is again universal homogeneous triangle-free in which the coloring takes no more than $T(G)$ colors. This is the first such result for a homogeneous structure omitting copies of some non-trivial finite structure. The proof entails developments of new broadscale techniques, including a flexible method for constructing trees which code $\mathcal{H}_3$ and the development of their Ramsey theory.