Two questions. 1. Do the unit length edges you add have to be horizontal or vertical? 2. How is the first edge added for a given seed? Should the seed be one of its endpoints or should it just lie on it somewhere?

Awesome. This works, thanks! Can you give me some idea about how you came up with this by the way?

I am confused. Doesn't the second figure show that we can do 10 points?

I am talking about the generalization of the Czaszar polyhedron. It's a polyhedron whose skeleton is $K_7$. For n = 6, 7, 8, 9, 10, 11, there cannot be a polyhedron whose skeleton is $K_n$, which can be shown using the Euler characteristic. So the next candidate is $K_12$, and that's still open. I had assumed that 3-polytopes are the same as polyhedra. But it seems they are the same as "convex" polyhedra?

I meant $K_{12}$.

I have heard that it's still not known if the complete graph on 12 vertices can be realized as the 1-skeleton of a 3-polytope. But according to what you have said, it should have been decidable. So can't we just use Tarski's algorithm once on $K_12$ and check? Or is the input size really really big for this case?

Then, do we know of a theorem that characterizes the 1-skeleton of some other kind of polytopes (i.e., not necessarily convex)?

I wonder what he meant by "similar theorem". Did he mean that a theorem characterizing the 1-skeleton of convex polytopes is not known or that a theorem characterizing the 1-skeleton of general polytopes is not known?