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

Suppose there is a counterexample to the hf/hf problem. Is the set of all hyperfinite equivalence relations $\mathbf{\Sigma}_21$-complete?

— Open Problems in Mathematical Logic  (2608.26628 - Barmpalias et al., 27 Aug 2026) in 2025 Section 12