Embedding into Rectilinear Spaces

We show that the problem whether a given finite metric space (X, d) can be embedded into the rectilinear space R m can be formulated in terms of m-colorability of a certain hypergraph associated with (X, d). This is used to close a gap in the proof of an assertion of Bandelt and Chepoi [2] on certain critical metric spaces for this embedding problem. We also consider the question of determining the maximum number of equidistant points that can be placed in the m-dimensional rectilinear space and show that this number is equal to 2m for m ≤ 3.