Big Ramsey degrees and the two-branching pseudotree

Abstract: We prove that each finite chain in the two-branching countable ultrahomogeneous pseudotree has finite big Ramsey degrees. This is in contrast to the recent result of Chodounsk\'{y}, Eskew, and Weinert that antichains of size two have infinite big Ramsey degree in the pseudotree. Combining a lower bound result of theirs with work in this paper shows that chains of length two in the pseudotree have big Ramsey degree exactly seven. The pseudotree is the first example of a countable ultrahomogeneous structure in a finite language in which some finite substructures have finite big Ramsey degrees while others have infinite big Ramsey degrees.