Complexity of measurable chromatic-number codes

Characterize the descriptive complexity, for fixed k≥3 and a fixed Borel measure μ on a standard Borel space X, of the codes for locally countable Borel graphs G satisfying χ_μ(G)≤k, and determine whether measurable 2-colorability of acyclic bounded-degree graphings has strictly lower complexity than measurable 3-colorability.

Background

Unlike the Borel setting, the paper notes that little is known about projective complexity in the measurable setting. It asks both for the complexity of the coding sets and for a possible measurable analogue of the distinction between 2-colorability and k-colorability for k≥3.

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 immediately following the hyperfinite-graph problem, Section 6 (Complexity)

Let $k\ge 3$ and fix a Borel measure $\mu$ on some standard Borel space $X$. 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 11 (Open problems), Complexity