Not and answer, but a long comment:   I don't think a uniform rectangular grid works.  Take a planar poisson process with intensity 1 and look at   (0,n)x(0,n).  This is your setup exept I have replaced the uniforms with a poisson process with an expected n^2 events in the space. Divie it  into a rectangular grid with (cn)^2  rectangles of area 1/c^2 each.    The  number of points in each rectangle is poisson with parameter 1/c^2, and you wonder if any have of the (cn)^2 rectangles have two points in them.  As the parameter is fixed there is a probability going to 1 that at least one of them does.