# Number of planar graphs, given an embedding

I want to find an upper bound on the number of planar graphs with $$n$$ vertices, assuming that we are given some embedding for those vertices beforehand. In particular, Im interested in either showing for every embedding there is $$2^{o(n\log(n))}$$ planar graphs, or showing that there exists an embedding with $$2^{\Omega(n\log(n))}$$ different planar graphs.

I know that without fixing embedding there is $$2^{\Theta(n\log(n))}$$, but my question differs from it since we fix an embedding beforehand.

I believe that still there is an embedding with $$2^{\Omega(n\log(n))}$$, but I couldn't prove or disprove it. Here is an attempt I made to try and calculate the number of planar graphs for some embedding I believed would be "hard", but it didn't work out as expected.

Pach and Wenger proved in their paper Embedding planar graphs at fixed vertex locations that if $$p_1,\ldots,p_n$$ are $$n$$ different points on the plane, then every planar graph on $$n$$ vertices $$v_1,\ldots,v_n$$ has a planar embedding in which $$v_i$$ is found at position $$p_i$$.