The keyword for searching is "infinite exponent partition relation."
The following article, for example, has many interesting results concerning the infinite exponent partition relation $\omega\to(\omega)^\omega$, which asserts that every coloring function has your Ramsey property, and its connection with weak forms of choice.
Kleinberg, E. M.; Seiferas, J. I., Infinite exponent partition relations and well-ordered choice, J. Symb. Log. 38, 299-308 (1973). ZBL0274.04004.
In particular, they prove that under well-ordered choice, which is a weakening of the axiom of choice to well-ordered index sets (e.g. countable choice is an instance of this), then case $\omega\to(\omega)^\omega$ is the only possible case.
Meanwhile, it is mentioned there that Andrian Mathias, in his dissertation, proved that the infinite exponent partition relation $\omega\to(\omega)^\omega$ is consistent with countable choice and indeed with dependent choice.