  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.
