Dushnik-Miller dimension of TD-Delaunay complexes
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TD-Delaunay graphs, where TD stands for triangular distance, is a variation of the classical Delaunay triangulations obtained from a specific convex distance function. Bonichon et. al. noticed that every triangulation is the TD-Delaunay graph of a set of points in R^2, and conversely every TD-Delaunay graph is planar. It seems natural to study the generalization of this property in higher dimensions. Such a generalization is obtained by defining an analogue of the triangular distance for R^d. It is easy to see that TD-Delaunay complexes of R^d-1 are of Dushnik-Miller dimension d. The converse holds for d=2 or 3 and it was conjectured independently by Mary and Evans et. al. to hold for larger d. Here we disprove the conjecture already for d = 4.
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