Big Ramsey degrees in universal inverse limit structures

TL;DR

This paper extends topological Ramsey space constructions to inverse limit structures of Fra"{}sse9 classes, proving finite big Ramsey degrees for these structures under certain conditions.

Contribution

It introduces new topological Ramsey spaces for inverse limit structures and establishes finite big Ramsey degrees for a broad class of Fra"{}sse9 classes, including exact degrees in specific cases.

Findings

Universal inverse limit structures have finite big Ramsey degrees.

Construction of topological Ramsey spaces extends previous work to new classes.

Exact big Ramsey degrees characterized for free amalgamation classes, tournaments, and partial orders.

Abstract

We build a collection of topological Ramsey spaces of trees giving rise to universal inverse limit structures,extending Zheng's work for the profinite graph to the setting of Fra\"{\i}ss\'{e} classes of finite ordered binary relational structures with the Ramsey property. This work is based on the Halpern-L\"{a}uchli theorem, but different from the Milliken space of strong subtrees. Based on these topological Ramsey spaces and the work of Huber-Geschke-Kojman on inverse limits of finite ordered graphs, we prove that for each such Fra\"{\i}ss\'{e} class, its universal inverse limit structure has finite big Ramsey degrees under finite Baire-measurable colorings. For such \Fraisse\ classes satisfying free amalgamation as well as finite ordered tournaments and finite partial orders with a linear extension, we characterize the exact big Ramsey degrees.