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

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Book cover Computational Discrete Mathematics

Part of the book series: Lecture Notes in Computer Science ((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.

Supported by a DFG Gerhard-Hess-Forschungsförderungspreis (Zi 475/2-3).

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Authors and Affiliations

  1. MA 7-1, Dept. Mathematics, TU Berlin, D-10623, Berlin, Germany

    Günter M. Ziegler

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  1. Günter M. Ziegler
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Editors and Affiliations

  1. Institut für Informatik, Freie Universität Berlin, Takustr. 9, 14195, Berlin, Germany

    Helmut Alt

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Ziegler, G.M. (2001). Coloring Hamming Graphs, Optimal Binary Codes, and the 0/1-Borsuk Problem in Low Dimensions. In: Alt, H. (eds) Computational Discrete Mathematics. Lecture Notes in Computer Science, vol 2122. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45506-X_12

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  • DOI: https://doi.org/10.1007/3-540-45506-X_12

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42775-9

  • Online ISBN: 978-3-540-45506-6

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