Extensions
This theorem has been generalized by Brightwell and Trotter (1993, 1997) to a tight bound on the dimension of the height-three partially ordered sets formed analogously from the vertices, edges and faces of a convex polyhedron, or more generally of an embedded planar graph: in both cases, the order dimension of the poset is at most four. However, this result cannot be generalized to higher-dimensional convex polytopes, as there exist four-dimensional polytopes whose face lattices have unbounded order dimension.
Even more generally, for abstract simplicial complexes, the order dimension of the face poset of the complex is at most 1 + d, where d is the minimum dimension of a Euclidean space in which the complex has a geometric realization (Ossona de Mendez 1999, 2002).
Read more about this topic: Schnyder's Theorem
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