Big Ramsey degrees and infinite languages

Samuel Braunfeld, David Chodounský, Noé de Rancourt, Jan Hubička, Jamal Kawach, Matěj Konečný

Published 2023-01-30Version 1

Despite significant progress in the study of big Ramsey degrees, the big Ramsey degrees of many classes of structures with finite small Ramsey degrees remain unknown. In this paper, we investigate the big Ramsey degrees of unrestricted relational structures in (possibly) infinite languages and demonstrate that they have finite big Ramsey degrees if and only if there are only finitely many relations of every arity. This is the first time that the finiteness of big Ramsey degrees has been established for an infinite-language random structure.