Ramsey properties of trees and $C$-relations via the partite construction

Source abstract

Finite convexly ordered binary branching $C$-relations have been viewed as a notable exception to the broad applicability of the partite construction in structural Ramsey theory. The absence of a proof of their Ramsey property by the partite construction was noted by Bodirsky in his 2015 survey and pointed out again by Hubička and Konečný in their 2026 survey. We overcome this limitation by giving proofs based on the partite construction of the known Ramsey properties of the class $\mathcal{CT}$ of finite convexly ordered treeable meet semilattices and of the classes of finite convexly ordered $C$-relations with arbitrary branching and with binary branching. Our proofs do not rely on Milliken's tree theorem. The key is to expand $\mathcal{CT}$ by level predicates and a predecessor function so that the locally finite subclass method applies. We then transfer the resulting Ramsey witnesses to the corresponding reduct classes and to the classes of convexly ordered $C$-relations.