On embeddings of CAT(0) cube complexes into products of trees via colouring their hyperplanes

We prove that the contact graph of a 2-dimensional CAT(0) cube complex X of maximum degree Δ can be coloured with at most ϵ(Δ)=MΔ26 colours, for a fixed constant M. This implies that X (and the associated median graph) isometrically embeds in the Cartesian product of at most ϵ(Δ) trees, and that the event structure whose domain is X admits a nice labelling with ϵ(Δ) labels. On the other hand, we present an example of a 5-dimensional CAT(0) cube complex with uniformly bounded degrees of 0-cubes which cannot be embedded into a Cartesian product of a finite number of trees. This answers in the negative a question raised independently by F. Haglund, G. Niblo, M. Sageev, and the first author of this paper.