Big Ramsey degrees of 3-uniform hypergraphs are finite

TL;DR

This paper proves that the universal homogeneous 3-uniform hypergraph has finite big Ramsey degrees, marking the first such result for structures in a non-binary language, and introduces a method to extend binary results to higher arities.

Contribution

It establishes the finiteness of big Ramsey degrees for 3-uniform hypergraphs and develops a general approach to extend binary relational results to higher arities.

Findings

Finite big Ramsey degrees for the universal homogeneous 3-uniform hypergraph

First non-binary structure with known finite big Ramsey degrees

A new method based on Milliken's Tree Theorem for higher arities

Abstract

We prove that the universal homogeneous 3-uniform hypergraph has finite big Ramsey degrees. This is the first case where big Ramsey degrees are known to be finite for structures in a non-binary language. Our proof is based on the vector (or product) form of Milliken's Tree Theorem and demonstrates a general method to carry existing results on structures in binary relational languages to higher arities.