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. Consequently, if ZF is consistent, those choice principles are insufficient to produce a non-Ramsey function.

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 Adrian 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. Consequently, if ZF is consistent, those choice principles are insufficient to produce a non-Ramsey function.

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.

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. Consequently, if ZF is consistent, those choice principles are insufficient to produce a non-Ramsey function.