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 ${\sf (ZF)}$?