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

@misc{lee2026ramsey,
  title = {{Ramsey properties of trees and $C$-relations via the partite construction}},
  author = {Hyoyoon Lee},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.18402},
  eprint = {2610.18402},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.18402},
}

TY  - UNPB
TI  - Ramsey properties of trees and C-relations via the partite construction
AU  - Hyoyoon Lee
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.18402
UR  - https://arxiv.org/abs/2610.18402
AB  - 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 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 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.
ER  -