A survey on big Ramsey structures

TL;DR

This survey consolidates recent progress in structural Ramsey theory, focusing on big Ramsey structures, their properties, and how they encode finite big Ramsey degrees across various examples.

Contribution

It provides a unified framework for understanding big Ramsey structures and shows how standard methods naturally produce such structures with desirable features.

Findings

Standard proof methods yield big Ramsey structures automatically.

Many examples fit into the proposed framework.

Expanding structures with unary functions affects big Ramsey degrees.

Abstract

In recent years, there has been much progress in the field of structural Ramsey theory, in particular in the study of big Ramsey degrees. In all known examples of infinite structures with finite big Ramsey degrees, there is in fact a single expansion of the structure, called a big Ramsey structure, which correctly encodes the exact big Ramsey degrees of every finite substructure simultaneously. The first half of the article collects facts about this phenomenon that have appeared in the literature into a single cohesive framework, thus offering a conceptual survey of big Ramsey structures. We present some original results indicating that the standard methods of proving finite big Ramsey degrees automatically yield big Ramsey structures, often with desirable extra properties. The second half of the article is a survey in the more traditional sense, discussing numerous examples from the literature and showing how they fit into our framework. We also present some general results on how big Ramsey degrees are affected by expanding structures with unary functions.