Corrigendum and Addendum to "Fra\"{\i}ss\'{e}'s Conjecture and big Ramsey degrees of structures admitting finite monomorphic decomposition''

TL;DR

This corrigendum corrects a misidentification in a previous paper regarding the reduct of the generic permutation, and demonstrates that the framework can establish finite big Ramsey degrees for new classes of structures, opening new research directions.

Contribution

It corrects a key error about the generic 2-dimensional partial order and shows the framework's applicability to new classes of structures with finite big Ramsey degrees.

Findings

The reduct is the generic 2-dimensional partial order, not the generic partial order.

The framework can establish finite big Ramsey degrees for previously unexplored classes.

The correction leads to new insights into the existence of finite big Ramsey degrees for certain relational structures.

Abstract

In Section 6 of the paper ``Fra\"{\i}ss\'{e}'s Conjecture and big Ramsey degrees of structures admitting finite monomorphic decomposition'', we applied the methods developed in earlier sections to show that a certain reduct of the generic permutation has finite big Ramsey degrees. Unfortunately, this reduct was incorrectly identified as the generic partial order. We are grateful to Jan Hubi\v{c}ka for bringing this error to our attention. In this note we correct the statements that rely on this misidentification and demonstrate that the reduct in question is in fact the generic 2-dimensional partial order. We emphasize that the arguments presented in Section 6 remain valid, with the sole exception of the Claim in the proof of Theorem 6.4, whose role was to (incorrectly) identify the reduct of the generic permutation as the generic partial order. This correction has an unexpected positive consequence. Rather than reproving a well-known result whose existing proof is already notably elegant, this note demonstrates that our general framework can be used to establish that a previously unexplored class of generic relational structures has finite big Ramsey degrees. This observation opens a potentially new direction for further research in the thriving area of big Ramsey combinatorics. In the addendum, we combine a recent result by Oudrar and Pouzet with our analysis of finite big Ramsey degrees for structures admitting finite monomorphic decomposition to characterize the existence of finite Big Ramsey degrees for all countable relational structures whose language has a linear order and age has polynomial growth.