On the Borsuk number of four-dimensional sets
Metric Geometry
2010-08-12 v2
Abstract
Borsuk conjectured that every n-dimensional bounded set of positive diameter can be partitioned into n+1 sets of smaller diameters. This conjecture was proved for n=2 by Borsuk, for n=3 first by Eggleston, and disproved for n > 297 by Hinrichs and Richer. It is not known if the conjecture holds for 3 < n < 298. The best upper bound for the number of subsets of smaller diameters a four-dimensional set can be partitioned into is nine. This estimate was given by Lassak in 1982. In this note we improve this estimate by one.
Cite
@article{arxiv.1007.2518,
title = {On the Borsuk number of four-dimensional sets},
author = {Zsolt Langi},
journal= {arXiv preprint arXiv:1007.2518},
year = {2010}
}
Comments
This paper has been withdrawn by the author due to a crucial computational error in the proof of the lemma