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. 

<cite authors="Kleinberg, E. M.; Seiferas, J. I.">_Kleinberg, E. M.; Seiferas, J. I._, [**Infinite exponent partition relations and well-ordered choice**](https://doi.org/10.2307/2272066), J. Symb. Log. 38, 299-308 (1973). [ZBL0274.04004](https://zbmath.org/?q=an:0274.04004).</cite>

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.