Empty Rectangles and Graph Dimension

TL;DR

This paper investigates the maximum edges in rectangle graphs and related graph classes, revealing exact bounds and their geometric and combinatorial interpretations, including connections to graph dimension and complex structures.

Contribution

It establishes exact maximum edge counts for rectangle graphs and their higher-dimensional analogs, linking geometric configurations to graph dimension and combinatorial complexes.

Findings

Maximum edges in rectangle graphs is 1/4 n^2 + n - 2.

Maximum edges in 3D box graphs is 7/16 n^2 + o(n^2).

Extremal point sets for maximum edges coincide with those for empty rectangles.

Abstract

We consider rectangle graphs whose edges are defined by pairs of points in diagonally opposite corners of empty axis-aligned rectangles. The maximum number of edges of such a graph on $n$ points is shown to be 1/4 n^2 +n -2. This number also has other interpretations: * It is the maximum number of edges of a graph of dimension $\bbetween{3}{4}$, i.e., of a graph with a realizer of the form $\pi_1,\pi_2,\ol{\pi_1},\ol{\pi_2}$. * It is the number of 1-faces in a special Scarf complex. The last of these interpretations allows to deduce the maximum number of empty axis-aligned rectangles spanned by 4-element subsets of a set of $n$ points. Moreover, it follows that the extremal point sets for the two problems coincide. We investigate the maximum number of of edges of a graph of dimension $\between{3}{4}$, i.e., of a graph with a realizer of the form $\pi_1,\pi_2,\pi_3,\ol{\pi_3}$. This maximum is shown to be $1/4 n^2 + O(n)$. Box graphs are defined as the 3-dimensional analog of rectangle graphs. The maximum number of edges of such a graph on $n$ points is shown to be $7/16 n^2 + o(n^2)$.