I am working on the Banach–Tarski paradox and the fact that the Hahn–Banach theorem implies that paradox. The proof involves the equivalence of the Hahn–Banach theorem and the fact that for every Boolean algebra $\mathcal A$ there is a finitely additive "measure" $\mathrm m:\mathcal A \to [0,+\infty]$ such that some element of $\mathcal A$ has finite measure. $\mathsf{ZF}$ is not enough to derive the Banach-Tarski paradox, nor is $\mathsf{ZF}+\mathsf{DC}$.

However, I am interested in reversing the roles, in order to characterize the Banach–Tarski paradox in terms of choice-like axioms. Hahn–Banach is strictly weaker than the axiom of choice or the ultrafilter lemma/boolean prime ideal theorem.

Can we find a weaker axiom to derive the Banach–Tarski paradox? Do you know any reference or have any ideas on this? Are there weaker axioms that should suffice in order to get the Banach–Tarski paradox? Is it possible to reformulate the Banach–Tarski paradox in a way that this assumption allows us in $\mathsf{ZF}$ to recover the Hahn–Banach extension theorem?

These questions do not seem clear to me, so I would be very pleased if they are somehow absurd. Thanks in advance.

Consequences of the Axiom of Choice? That's where people always send me whenever I ask a question like this. $\endgroup$5more comments