For any set $A\neq\emptyset$, denote by $[A]^A$ the collection of sets $B\subseteq A$ such that there is a bijection $\varphi:B\to A$. If ${\cal S}\subseteq [A]^A$, we say that $B\in[A]^A$ is *monochromatic* with respect to ${\cal S}$ if either 

1. ${\cal P}(B)\cap {\cal S} = \emptyset$, or
2. $\big({\cal P}(B)\cap[A]^A\big) \subseteq {\cal S}$.

Consider the following statement in ${\sf (ZF)}$:

> (Non-Mono) If $A$ is an infinite set, then there is ${\cal S}\subseteq [A]^A$ such that no $B\in[A]^A$ is monochromatic with respect to ${\cal S}$.

(Non-Mono) is false for finite sets $A$ because $[A]^A = \{A\}$. On the other hand, (Non-Mono) is a [theorem](https://dominiczypen.wordpress.com/2023/07/18/non-ramsey-functions-property-b-and-the-axiom-of-choice/) of ${\sf (ZF) + (AC)}$. 

**Questions.**
1) Does (Non-Mono) imply the Axiom of Choice in ${\sf (ZF)}$?
2) If not, does (Non-Mono) imply  the [Partition Priniple](https://mathoverflow.net/questions/420280/does-the-partition-principle-imply-dc) or the weaker [Dual Cantor-Bernstein theorem](https://mathoverflow.net/questions/420280/does-the-partition-principle-imply-dc)?