2009/09/02 by Kyle Kinneberg, Aaron Mazel-Gee, Kinneberg, Kyle E. +5
Computer Science · Mathematics · #54H25 #55M20 #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT) #Limits and Structures in Graph Theory #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.0909.0471
openalex publication_date 2009/09/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.