Dushnik-Miller dimension of d-dimensional tilings with boxes

TL;DR

This paper generalizes the relationship between intersection graphs and Dushnik-Miller dimension from planar rectangles to higher-dimensional boxes, establishing an upper bound on the dimension based on the number of intersecting boxes.

Contribution

It extends known planar results to $R^d$, proving that tilings with at most $d+1$ intersecting boxes have Dushnik-Miller dimension at most $d+1$.

Findings

Dushnik-Miller dimension of intersection graphs is at most three for planar graphs.

Generalization to $R^d$ shows the dimension is at most $d+1$ for certain tilings.

Provides a link between geometric tilings and poset dimension theory.

Abstract

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 $R^d$ by considering tilings of $R^d$ 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 $d+1$.