Suppose we are given a set of $n$ points on the plane. How many different planar graphs can we form on those $n$ vertices, assuming that each edge must form a straight line between the two vertices it connects?
I'm interested in either showing for every set of points there are $2^{o(n\log(n))}$ planar graphs; or showing that there exists a set of points with $2^{\Omega(n\log(n))}$ different planar graphs.
I recently asked a similar question without the constraints that the edges form a straight line, and the answer I got directly utilizes this fact. I didn't specify this in the previous question, but I am actually interested only in graphs with those constraints. So, if anyone knows where I can find material about the number of such graphs, I would be glad to know about it as well.
Thanks in advance!
