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Coloring Hamming Graphs, Optimal Binary Codes, and the 0/1-Borsuk Problem in Low Dimensions

  • Günter M. Ziegler
Chapter
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2122)

Abstract

The 0/1-Borsuk problem asks whether every subset of {0, 1}d can be partitioned into at most d + 1 sets of smaller diameter. This is known to be false in high dimensions (in particular for d ≥ 561, due to Kahn & Kalai, Nilli, and Raigorodskii), and yields the known counter-examples to Borsuk’s problem posed in 1933.

Here we ask whether there might be counterexamples in low dimension as well. We show that there is no counterexample to the 0/1-Borsuk conjecture in dimensions d ≤ 9. (In contrast, the general Borsuk conjecture is open even for d = 4.)

Our study relates the 0/1-case of Borsuk’s problem to the coloring problem for the Hamming graphs, to the geometry of a Hamming code, as well as to some upper bounds for the sizes of binary codes.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Günter M. Ziegler
    • 1
  1. 1.MA 7-1, Dept. MathematicsTU BerlinBerlinGermany

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