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 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 or the weaker Dual Cantor-Bernstein theorem?