This paper investigates big Ramsey degrees of unrestricted relational
structures in (possibly) infinite languages. While significant progress has
been made in studying big Ramsey degrees, many classes of structures with
finite small Ramsey degrees still lack an understanding of their big Ramsey
degrees. 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 that finiteness of big Ramsey degrees has been established for an
infinite-language random structure. Our results represent an important step
towards a better understanding of big Ramsey degrees for structures with
relations of arity greater than two.Comment: 21 pages. An updated version strengthening the statement of the
positive results and fixing a mistake in the earlier version of the negative
result which now needs an extra assumptio