Let $[\omega]^\omega$ denote the collection of infinite subsets of $\omega$, and let $c:[\omega]^\omega\to\{0,1\}$ be a function. We say that $a\in [\omega]^\omega$ is **monochromatic** with respect to $c$ if the restriction $c|\_{{\cal P}(a)\cap [\omega]^\omega}$ of $c$ to the infinite subsets of $a$ is constant. Moreover, $c$ is said to be **Ramsey** if there is $a\in [\omega]^\omega$ such that $a$ is monochromatic with respect to $c$.


Using the [Axiom of Choice](https://en.wikipedia.org/wiki/Axiom_of_choice) it is possible to construct [a non-Ramsey function $c:[\omega]^\omega\to\{0,1\}$](https://dominiczypen.wordpress.com/2023/07/18/non-ramsey-functions-property-b-and-the-axiom-of-choice/).


**Question.** Does the existence of a non-Ramsey function $c:[\omega]^\omega\to\{0,1\}$ imply some (weak) form of (AC) that cannot be proved in ${\sf (ZF)}$?



Non-Ramsey functions $c:[\omega]^\omega\to\{0,1\}$ and the Axiom of 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**](https://doi.org/10.2307/2272066), J. Symb. Log. 38, 299-308 (1973). [ZBL0274.04004](https://zbmath.org/?q=an:0274.04004).


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.


