Equivalence of Nielsen–Schreier to the axiom of choice

Establish whether the Nielsen–Schreier Theorem (every subgroup of a free group is free) is equivalent to the axiom of choice in ZF set theory; that is, resolve if Nielsen–Schreier can be proved in ZF precisely when choice holds.

Background

Schreier’s proof uses choice‑like principles; Levi and subsequent authors noted reliance on well‑ordering. Intensive set‑theoretic investigations have explored the exact metamathematical strength of Nielsen–Schreier within ZF.

The authors highlight that, despite significant progress, a complete equivalence characterization remains unproven.