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

Dushnik-Miller dimension of d-dimensional tilings with boxes