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THE 0/1-BORSUK CONJECTURE IS GENERICALLY TRUE FOR EACH FIXED DIAMETER
 

Summary: THE 0/1-BORSUK CONJECTURE IS GENERICALLY
TRUE FOR EACH FIXED DIAMETER
JONATHAN P. MCCAMMOND AND G¨UNTER ZIEGLER
Abstract. In 1933 Karol Borsuk asked whether every compact subset of Rd
can be decomposed into d + 1 subsets of strictly smaller diameter. The 0/1-
Borsuk conjecture asks a similar question using subsets of the vertices of a d-
dimensional cube. Although counterexamples to both conjectures are known,
we show in this article that the 0/1-Borsuk conjecture is true when d is much
larger than the diameter of the subset of vertices. In particular, for every k,
there is a constant n which depends only on k such that for all configurations
of dimension d > n and diameter 2k, the set can be partitioned into d - 2k + 2
subsets of strictly smaller diameter. Finally, L´asl´o Lov´asz's theorem about the
chromatic number of Kneser's graphs shows that this bound is in fact sharp.
Contents
1. Introduction 1
2. Configurations 2
3. Borsuk graphs 3
4. Shells 4
5. Kneser graphs 4
6. Central sets 5

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics