  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 $7/16 n^2
+ o(n^2)$.
