Coloring Hamming graphs, optimal binary codes, and the 0/1-Borsuk problem in low dimensions

Authors:
Günter M. Ziegler
Affiliations:
TU Berlin, Berlin, Germany
Venue:
Computational Discrete Mathematics
Year:
2001

Citing 4
Cited 4

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Abstract

The o/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 counterexamples 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.