Dushnik-Miller dimension of stair contact complexes

Abstract

The theorem of Schnyder asserts that a graph is planar if and only if the Dushnik-Miller dimension of its poset of incidence is at most 3. Trotter aksed how this can be generalized to higher dimensions. Towards this goal, Dushnik-Miller dimension has been studied in terms of TD-Delaunay complexes, in terms of orthogonal surfaces, and in terms of polytopes. Here we consider the relation between the Dushnik-Miller dimension and contact systems of stairs in Rd.We propose two different definitions of stairs in Rd which are connected to the Dushnik-Miller dimension as follows.  The first definition allows us to characterize supremum sections, which are simplicial complexes related to the Dushnik-Miller dimension, in two different ways.  The second definition provides for any Dushnik-Miller dimension at most d+1 complex a representation as a contact system of stairs in Rd.