  We show that the universal homogeneous partial order has finite big Ramsey
degrees, and we discuss several corollaries. Our proof relies on parameter
spaces and the Carlson--Simpson theorem rather than (a strengthening of) the
Halpern--L\"auchli theorem and the Milliken tree theorem, which are typically
used to bound big Ramsey degrees in existing literature (originating from 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 indivisibility (vertex partition) result of Nguyen van Th\'e and
Sauer, we give 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.
