I have the following exercise (ex. 9.15 taken from here):
Let $M$ be an arbitrary structure. Prove that $M$ has an $\omega$-homogeneous elementary extension of the same cardinality. (There is no assumption on the cardinality of the language).
A structure $N$ is said (in this case, I don't think it is standard notation) to be $\omega$-homogeneous if every elementary map between finitely generated substructures of $N$ extends to an automorphism.
I think I have proved the result for $|L|\le|M|$ ($L$ is the f.o. language we are working with) though in a rather long and complicated way. But I have absolutely no idea on how to prove the result when $|L|> |M|$ since I cannot invoke Löwenheim–Skolem without losing control on the cardinality of the elementary extensions I'd use. Any hint?
Thanks!