Complexity of measurable colorability

Determine the descriptive complexity of codes for locally countable Borel graphs whose μ-measurable chromatic number is at most k, and characterize whether measurable 2-colorability of acyclic bounded-degree graphings has lower complexity than measurable 3-colorability.

Background

The paper notes that complexity results for measurable colorability are largely absent, although straightforward upper bounds are available. It asks for a precise complexity classification and a possible separation between two- and three-colorability.

References

What is the complexity of the codes of locally countable Borel graphs ${G}$ with $\chi_\mu({G}) \leq k$? Is there a precise way in which deciding the measurable $2$-colorability of an acyclic bounded degree graphing is easier than to decide its $3$-colorability?

— From descriptive to distributed  (2502.15347 - Grebík et al., 21 Feb 2025) in Problem 12, Section 7 (Complexity)