Complexity of hyperfinite acyclic graphs with Δ-colorings

Determine whether the class of Δ-regular acyclic hyperfinite Borel graphs admitting a Borel vertex coloring with Δ colors is Σ¹₂-complete for every Δ≥3.

Background

The paper records that Borel Δ-colorability is highly complex even for Δ-regular acyclic graphs, while hyperfiniteness is itself tied to major unresolved questions in descriptive set theory. It asks whether imposing hyperfiniteness preserves Σ¹₂-completeness for the Δ-colorability class.

References

Do $\Delta$-regular acyclic Borel hyperfinite Borel graphs that have Borel chromatic number at most $\Delta$ form a $\Sigma1_2$-complete?

— From descriptive to distributed  (2502.15347 - Grebík et al., 21 Feb 2025) in Problem immediately following the discussion of Borel hyperfinite graphs, Section 6 (Complexity)

Let $\Delta\ge 3$. Do $\Delta$-regular acyclic Borel hyperfinite Borel graphs that have Borel chromatic number at most $\Delta$ form a $\Sigma1_2$-complete?

— From descriptive to distributed  (2502.15347 - Grebík et al., 21 Feb 2025) in Problem 11, Section 11 (Open problems), Complexity