Big Ramsey degrees using parameter spaces

TL;DR

This paper demonstrates that the universal homogeneous partial order and other structures have finite big Ramsey degrees using parameter spaces and the Carlson-Simpson theorem, offering new proofs and broader applications.

Contribution

It introduces a novel technique employing parameter spaces and the Carlson-Simpson theorem to establish finite big Ramsey degrees, replacing traditional methods like Halpern-L"auchli and Milliken theorems.

Findings

Universal homogeneous partial order has finite big Ramsey degrees.

Finite big Ramsey degrees for the homogeneous universal triangle-free graph.

Upper bounds on big Ramsey degrees of metric spaces with finitely many distances.

Abstract

We show that the universal homogeneous partial order has finite big Ramsey degrees and discuss several corollaries. Our proof relies on parameter spaces and the Carlson-Simpson theorem rather than on (a strengthening of) the Halpern-L\"auchli theorem and the Milliken tree theorem, which are typically used to bound big Ramsey degrees in the existing literature (originating from the work of Laver and Milliken). This new technique has many additional applications. We show that the homogeneous universal triangle-free graph has finite big Ramsey degrees, providing a short proof of a recent result by Dobrinen. Moreover, generalizing an indivisibility (vertex partition) result of Nguyen van Th\'e and Sauer, we give an upper bound on big Ramsey degrees of metric spaces with finitely many distances. This leads to a new combinatorial argument for the oscillation stability of the Urysohn Sphere.