This paper has an interesting story. The result was obtained
simultaneously by two "teams" which independently
worked over the problem without knowing about the other team.
The both teams reduced the problem to the key "zigzag" lemma: *every
subspace E of R^{n} contain a zigzag of
order dim E*. However, the two teams came up with
completely different proofs of this lemma. Namely,
the proof via combinatorial
properties of polytopes was obtained by myself and my PhD student
A.Popov. The proof based on algebraic topology was obtained by
I.Chalendar, E.Fricain, and D.Timotin. After we learned about each
others' proofs we decided to combine our proofs into a joint paper.
Hence, the paper contains two completely different proofs of the
result. Finally, after the paper was published, we learned that
the