After a talk of mine Janos Pach pointed me to
to the following paper:

  N.Alon, Z.F\"uredi and M.Katchalski
  Separating pairs of points
  Europ. J. Comb. 6 (1985) 205-210.

They analyse the maximum number of edges of a graph of
empty boxes in dimension $d$.
For $d=2$ they prove that the max is
    $n^2/4+n-2$
and for $d>2$ they determine the asymptotics:
    $(1-\frac{1}{2^{2^{d-1}-1}})\frac{n^2}{2} + o(n^2)$.

This makes Theorem 1 and Theorem 3 of my paper obsolete.