On embeddings of CAT(0) cube complexes into products of trees

We prove that the contact graph of a 2-dimensional CAT(0) cube complex $\mathbf{X}$ of maximum degree $\mathrm{\Delta }$ can be coloured with at most $ϵ(\mathrm{\Delta })=M{\mathrm{\Delta }}^{26}$ colours, for a fixed constant $M$. This implies that $\mathbf{X}$ (and the associated median graph) isometrically embeds in the Cartesian product of at most $ϵ(\mathrm{\Delta })$ trees, and that the event structure whose domain is $\mathbf{X}$ admits a nice labeling with $ϵ(\mathrm{\Delta })$ 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.