  We prove crossing number inequalities for geometric graphs whose vertex sets
are taken from a d-dimensional grid of volume N and give applications of these
inequalities to counting the number of non-crossing geometric graphs that can
be drawn on such grids.
  In particular, we show that any geometric graph with m >= 8N edges and with
vertices on a 3D integer grid of volume N, has \Omega((m^2/n)\log(m/n))
crossings. In d-dimensions, with d >= 4, this bound becomes \Omega(m^2/n). We
provide matching upper bounds for all d. Finally, for d >= 4 the upper bound
implies that the maximum number of crossing-free geometric graphs with vertices
on some d-dimensional grid of volume N is n^\Theta(n). In 3 dimensions it
remains open to improve the trivial bounds, namely, the 2^\Omega(n) lower bound
and the n^O(n) upper bound.
