Borel versus Baire complexity on grids

Determine whether BOREL(Z^n) equals BAIRE(Z^n) for every integer n at least 2.

Background

The paper proves that BAIRE(Zn) and MEASURE(Zn) are distinct for n at least 2, but does not resolve the relationship between BOREL(Zn) and BAIRE(Zn). The unresolved equality is one of the remaining comparisons in the hierarchy of descriptive complexity classes for LCL problems on grids.

References

It remains an open problem to determine whether $BOREL(Zn) = BAIRE(Zn)$.

— Separating complexity classes of LCL problems on grids  (2501.17445 - Berlow et al., 29 Jan 2025) in Section 1, immediately after Theorem 1.2