Does the Partition Principle imply the Axiom of Choice?

• $\mathsf{ZF}$: Zermelo–Fraenkel set theory without the Axiom of Choice.
• Partition Principle ($\mathsf{PP}$): for sets $A, B$, if there is a surjection $A \twoheadrightarrow B$, then there is an injection $B \hookrightarrow A$.
• Axiom of Choice ($\mathsf{AC}$): every family of nonempty sets has a choice function.

• $\mathsf{PP}$ is known to imply several weaker choice-like consequences.
• $\mathsf{PP}$ has been shown equivalent to $\mathsf{AC}$ under various additional hypotheses, but no such additional hypothesis is known to be eliminable in general.
• The problem has resisted both a proof of the implication and the construction of a permutation/symmetric or forcing model separating $\mathsf{PP}$ from $\mathsf{AC}$.

This problem is notable for its longevity — it has remained open for over a century despite being one of the most natural questions one can ask about weak forms of choice, and despite extensive work relating $\mathsf{PP}$ to essentially every other known choice principle.

• Rubin, H. and Rubin, J. E. "Equivalents of the Axiom of Choice II." North-Holland, 1985 (catalogs known implications among choice principles, including this one).
• Halbeisen, L. "Combinatorial Set Theory: With a Gentle Introduction to Forcing." Springer, 2012 (survey of choice principles, including $\mathsf{PP}$).