A cubical antipodal theorem

Details A cubical antipodal theorem A cubical antipodal theorem

2009-09-02

arxiv.org/abs/0909.0471v1

Mathematics

Combinatorics

15 pages, 2 figures. See related work at http://www.math.hmc.edu/~su/papers.html

Scientific paper

The classical Lusternik-Schnirelman-Borsuk theorem states that if a d-sphere is covered by d+1 closed sets, then at least one of the sets must contain a pair of antipodal points. In this paper, we prove a combinatorial version of this theorem for hypercubes. It is not hard to show that for any cover of the facets of a d-cube by d sets of facets, at least one such set contains a pair of antipodal ridges. However, we show that for any cover of the ridges of a d-cube by d sets of ridges, at least one set must contain a pair of antipodal k-faces, and we determine the maximum k for which this must occur, for all dimensions except d=5.

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