Big Ramsey Degrees and Infinite Languages

TL;DR

This paper proves that certain infinite relational structures with finitely many relations of each arity have finite big Ramsey degrees, advancing understanding of these degrees in infinite languages and structures.

Contribution

It establishes the finiteness of big Ramsey degrees for structures with finitely many relations of each arity, including for the first time a random structure in an infinite language.

Findings

Finite big Ramsey degrees for structures with finitely many relations of each arity.

First proof of finiteness for a random structure in an infinite language.

Progress towards understanding big Ramsey degrees in structures with higher arity relations.

Abstract

This paper investigates big Ramsey degrees of unrestricted relational structures in (possibly) infinite languages. Despite significant progress in the study of big Ramsey degrees, the big Ramsey degrees of many classes of structures with finite small Ramsey degrees are still not well understood. We show that if there are only finitely many relations of every arity greater than one, then unrestricted relational structures have finite big Ramsey degrees, and give some evidence that this is tight. This is the first time finiteness of big Ramsey degrees has been established for a random structure in an infinite language. Our results represent an important step towards a better understanding of big Ramsey degrees for structures with relations of arity greater than two.