Motivation. The three-dimensional cube can be formalized by ${\cal P}(\{0,1,2\})$$\mathcal P(\{0,1,2\})$ where vertices $x,y\in{\cal P}(\{0,1,2\})$$x,y\in\mathcal 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)$$x\mathbin\Delta y := (x\setminus y) \cup (y\setminus x)$ contains exactly $1$ element. If we want to assign to each vertex one of the colors black and white such that no two vertices connected by an edge get the same color, we use the concept of parity: color any vertices containing an even number of elements black, and the rest white. This "parity algorithm" works for any ${\cal P}(X)$$\mathcal 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)$$\mathcal B\subseteq \mathcal P(X)$ such that whenever $a,b\in {\cal P}(X)$$a,b\in \mathcal P(X)$ with $a\Delta b = \{x\}$$a\mathbin\Delta b = \{x\}$ for some $x\in X$, then ${\cal B}$$\mathcal B$ contains exactly one of $a,b$$a$, $b$.
The Parity Principle is a theorem in ${\sf (ZFC)}$$(\textsf{ZFC})$. Does the Parity Principle imply the Axiom of Choice in ${\sf (ZF)}$$(\textsf{ZF})$?
