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Theorem (Van Kampen-Flores 1930s) From any 7 points in four-dimensional space one can choose two disjoint triples such that the triangles with vertices at the triples intersect each other.

This theorem has several beautiful folklore elementary proofs based on the following ideas:

1) the parity of the number of intersecting pairs of triangles does not depend on the choice of the 7 points;

2) reduction to the Conway-Gordon-Sachs theorem on 6 points in space by a projection paper link;

3) analysis of convex linear combinations of the 7 points in the spirit of the proof of the Radon theorem from convex geometry.

Do you know any references where the proofs based on these ideas (especially, the 3rd one) are published? Are there any other elementary proofs known?

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your third proof is on page 23 and following of Jiri Matousek's lecture notes.

an extended version of the lecture notes published by Springer contains two alternative proofs on pages 117 and 123

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  • $\begingroup$ By the third elementary proof I mean a one-paragraph long proof without references to any other results, really a gem! The proofs you mention use computation of deleted joins and Borsuk-Ulam theorem and are much more complicated. $\endgroup$
    – Mikhail Skopenkov
    Commented Mar 14, 2014 at 9:19

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