Choice strength of the dependence dichotomy

Determine which fragment of the Axiom of Choice is sufficient to prove the dependence dichotomy for functions on finite products, namely the assertion that every function satisfies either local essential dependence on at most one coordinate or the stated infinite two-block separation alternative.

Background

The paper proves a dichotomy for every function on a finite combinatorial cube: after finitely partitioning each coordinate space, the function locally depends on at most one coordinate on every resulting box, or there are two nonempty blocks of coordinates and sequences witnessing a strong separation of function values. The proof uses compactness of the Čech–Stone compactification, which in turn relies on the Axiom of Choice.

The paper shows that the dichotomy cannot be proved in ZF alone by constructing, from an infinite Dedekind-finite set, a function for which both alternatives fail. It also proves in ZF that the dichotomy holds when the coordinate spaces are well-orderable. The precise choice-theoretic strength required in the general case remains unresolved.