Big Ramsey Degrees and Infinite Languages
Samuel Braunfeld, David Chodounský, Noé de Rancourt, Jan Hubička, Jamal Kawach, Matěj Konečný
TL;DR
The paper addresses whether unrestricted relational structures in infinite languages can have finite big Ramsey degrees. It develops a framework based on Milliken's tree theorem, valuation trees, and envelope concepts to transfer colorings to valuation subtrees, enabling finiteness proofs. The main result shows finite big Ramsey degrees when there are finitely many relations of every arity greater than one, and it extends to unary relations, non-hypergraphs, and forbidding structures, marking a significant advance in understanding big Ramsey phenomena for higher-arity relations. The work also investigates the limits of these methods in infinite-branching settings, highlighting open questions about exact degrees and possible infinite big Ramsey degrees in broader languages.
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.