Measurable unfriendly coloring

Prove or disprove that every bounded-degree measured graph admits a measurable unfriendly coloring, and determine whether every bounded-degree Borel graph admits a Borel unfriendly coloring.

Background

An unfriendly coloring is a two-coloring in which every vertex has a color different from that of at least half of its neighbors. Partial measurable results require strong bounds on the Radon–Nikodym cocycle.

References

Does every measured graph of bounded degree admits a measurable unfriendly coloring? In fact, it is also open whether every Borel graph of bounded degree admits a Borel unfriendly coloring.

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

It also remains open whether finite average degree is necessary in [CT21, Theorem 1]: our obstruction admits no invariant probability.

— Unfriendly partitions of locally finite Borel graphs  (2609.11919 - Pelayo-Gómez, 10 Sep 2026) in Section 6.1, p. 12