The Hyperfinite-over-Hyperfinite Problem: Are hyperfinite-over-hyperfinite Borel equivalence relations hyperfinite?

Determine whether every hyperfinite-over-hyperfinite Borel equivalence relation—i.e., a Borel equivalence relation that admits a Borel assignment of linear orders on each class of type embeddable into the lexicographic order on Z×Z—is hyperfinite.

Background

Hyperfinite-over-hyperfinite equivalence relations are defined via the existence of a Borel Z^2-ordering (lexicographic order on Z×Z) on each class. This generalizes the classical characterization of hyperfiniteness via class-wise Z-orderings.

The paper proves that a sufficient condition—self-compatibility of such a Z^2-ordering—implies hyperfiniteness. However, in general it remains unknown whether every hyperfinite-over-hyperfinite relation must be hyperfinite.