The unfriendly partition conjecture for countable graphs

Let $G$ be a countable graph. An unfriendly bipartition of $G$ is a bipartition of $V(G)$ such that every vertex has at least as many neighbours in the other class as in its own.

Unfriendly partition conjecture. Every countable graph admits an unfriendly bipartition.

Every finite graph has an unfriendly bipartition, but the corresponding assertion for countably infinite graphs is a major open problem in infinite graph theory.