Dushnik-Miller dimension of contact systems of d-dimensional boxes

Planar graphs are the graphs with Dushnik-Miller dimension at most three (W. Schnyder, Planar graphs and poset dimension, Order 5, 323-343, 1989). Consider the intersection graph of interior disjoint axis-parallel rectangles in the plane. It is known that if at most three rectangles intersect on a point, then this intersection graph is planar, that is it has Dushnik-Miller dimension at most three. This paper aims at generalizing this from the plane to Rd by considering tilings of Rd with axis parallel boxes, where at most d+1 boxes intersect on a point. Such tilings induce simplicial complexes and we will show that those simplicial complexes have Dushnik-Miller dimension at most d+1.