Big Ramsey Degrees of the Universal Homogeneous Partial Order are finite

Abstract

We show that the universal homogeneous partial order has finite big Ramsey degrees. Our proof uses the Carlson-Simpson theorem rather than (a strengthening of) the Halpern-Läuchli and Milliken tree theorem which is the main tool used to give bounds on big Ramsey degrees.

We discuss two corollaries.

Keywords

• big Ramsey degree
• structural Ramsey theory
• homogeneous structure