Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A 64-dimensional two-distance counterexample to Borsuk's conjecture   In 1933 Karol Borsuk asked whether each bounded set in the n-dimensional
Euclidean space can be divided into n+1 parts of smaller diameter. The diameter
of a set is defined as the supremum (least upper bound) of the distances of
contained points. Implicitly, the whole set is assumed to contain at least two
points.
  The hypothesis that the answer to that question is positive became famous
under the name Borsuk's conjecture. Beginning with Jeff Kahn and Gil Kalai,
from 1993 to 2003 several authors have proved that in certain (almost all) high
dimensions such a division is not generally possible.
  In a paper published in 2013, Andriy V. Bondarenko constructed a
65-dimensional two-distance set of 416 vectors that cannot be divided into less
than 84 parts of smaller diameter. That was not just the first known
two-distance counterexample to Borsuk's conjecture but also a considerable
reduction of the lowest known dimension the conjecture fails in in general.
  This article presents a 64-dimensional subset of the vector set mentioned
above that cannot be divided into less than 71 (by A. Bondarenko 72) parts of
smaller diameter, that way delivering a two-distance counterexample to Borsuk's
conjecture in dimension 64.
  The contained proof relies on the results of some (combinatorial)
calculations. The additionally (in the source package) provided small computer
program G24CHK needs about one second for that task on a 1 GHz Intel PIII.
  Meanwhile a short paper by this author and Andries E. Brouwer that follows
the principal idea of this article but avoids the extensive computational part
has been submitted to The Electronic Journal of Combinatorics.

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The categories of this paper are: hep-th
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The categories of this paper are: cs
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The categories of this paper are: hep-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: cond-mat
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The categories of this paper are: bayes-an
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The categories of this paper are: nucl-th
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The categories of this paper are: atom-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: gr-qc
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The categories of this paper are: quant-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: solv-int
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The categories of this paper are: acc-phys
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The categories of this paper are: astro-ph
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The categories of this paper are: dg-ga