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 Rn 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 zigzag lemma appeared in J. Voigt, Pacific J. Maths 81 (1979), 1, 253-266.