Undecidability of hyperfiniteness complexity

Determine the descriptive complexity of the class of hyperfinite Borel graphs, in particular whether hyperfinite Borel graphs form a Σ¹₂ set.

Background

The paper calls the investigation of the complexity of hyperfinite Borel graphs a wide open question and notes its connection with major conjectures about hyperfiniteness.

References

A different, wide open question is the investigation of the complexity of hyperfinite Borel graphs, see, e.g., .

— From descriptive to distributed  (2502.15347 - Grebík et al., 21 Feb 2025) in Section 4, subsection on Decidability and complexity