It’s much easier than this. You can use a fun fact about a twisted cubic: no four distinct points on a twisted cubic are coplanar. So you take n points, draw a complete graph and use induction.
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In fact, this gives you not just an embedding, but one with all straight lines - so Fary’s Theorem is trivial for 3D graphs.
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you can show this using the "book embedding"! this person made some lovely illustrations of how this works:http://matan-matika.tumblr.com/post/123502746216/while-not-all-graphs-can-be-drawn-in-r2-every …
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