Big Ramsey degrees and the two-branching pseudotree
David Chodounský, Natasha Dobrinen, Thilo Weinert
TL;DR
This paper addresses finite big Ramsey degrees for finite chains in the countable two-branching pseudotree $\Psi$, contrasting with known infinite BRDs for certain substructures. The authors develop a coding-tree framework and prove a Halpern–Läuchli variant tailored to $\Psi$, together with the notion of almost antichains and diaries to control colorings of copies. They establish that every finite chain in $\Psi$ has finite big Ramsey degree, and that chains of length two have BRD exactly $7$, marking the first finite-language ultrahomogeneous structure with mixed finite/infinite BRD behavior among finite substructures. The results lay groundwork for topological Ramsey-space approaches to recover exact BRD values and suggest avenues for extending the theory to pseudotrees related to generalized Ważewski dendrites.
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 Chodounský, 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.