# The Ramsey Theory of Henson graphs

## TL;DR

This paper proves that for each $k extgreater 3$, the $k$-clique-free Henson graph has finite big Ramsey degrees, extending Ramsey theory to a new class of infinite, ultrahomogeneous graphs using novel coding techniques.

## Contribution

The paper introduces a new method of coding Henson graphs into strong coding trees and proves Ramsey theorems for these trees, establishing finite big Ramsey degrees for all $k extgreater 3$.

## Key findings

- Finite big Ramsey degrees are established for all $k extgreater 3$ Henson graphs.
- Development of strong coding trees as a new tool for Ramsey theory.
- Methodology opens pathways for studying big Ramsey degrees in other ultrahomogeneous structures.

## Abstract

Analogues of Ramsey's Theorem for infinite structures such as the rationals or the Rado graph have been known for some time. In this context, one looks for optimal bounds, called degrees, for the number of colors in an isomorphic substructure rather than one color, as that is often impossible. Such theorems for Henson graphs however remained elusive, due to lack of techniques for handling forbidden cliques. Building on the author's recent result for the triangle-free Henson graph, we prove that for each $k\ge 4$, the $k$-clique-free Henson graph has finite big Ramsey degrees, the appropriate analogue of Ramsey's Theorem. We develop a method for coding copies of Henson graphs into a new class of trees, called strong coding trees, and prove Ramsey theorems for these trees which are applied to deduce finite big Ramsey degrees. The approach here provides a general methodology opening further study of big Ramsey degrees for ultrahomogeneous structures. The results have bearing on topological dynamics via work of Kechris, Pestov, and Todorcevic and of Zucker.