**Motivation.** The *three-dimensional cube* can be formalized by ${\cal P}(\{0,1,2\})$ where vertices $x,y\in{\cal P}(\{0,1,2\})$ are connected by an edge if and only if their symmetric difference $x\Delta y := (x\setminus y) \cup (y\setminus x)$ contains exactly $1$ element. If we want to assign to each vertex one of the colors *black* and *white* such that no two vertices connected by an edge get the same color, we use the concept of *parity*: color any vertices containing an *even* number of elements *black*, and the rest *white*. This "parity algorithm" works for any ${\cal P}(X)$ where $X$ is a finite set. But what happens in the infinite case?

**Formal version.** Consider the following statement:

> (Parity principle:) If $X\neq \emptyset$ is a set, then there is ${\cal B}\subseteq {\cal P}(X)$ such that whenever $a,b\in {\cal P}(X)$ with $a\Delta b = \{x\}$ for some $x\in X$, then ${\cal B}$ contains exactly one of $a,b$. 

The Parity Principle is a theorem in ${\sf (ZFC)}$. Does the Parity Principle imply the Axiom of Choice in ${\sf (ZF)}$?