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 it is possible to construct a non-Ramsey function $c:[\omega]^\omega\to\{0,1\}$.

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

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 it is possible to construct a non-Ramsey function $c:[\omega]^\omega\to\{0,1\}$.

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 ${\sf (ZF)}$?