2010/07/15 by Zsolt Lángi, Zsolt Langi, Langi, Zsolt
Computer Science · Mathematics · Medicine · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #Medical and Biological Sciences #math.MG #msc:52A20 #msc:52A27 #msc:52A38
paper · pdf · doi:10.48550/arxiv.1007.2518
This paper has been withdrawn by the author due to a crucial computational error in the proof of the lemma
arxiv created 2010/08/11 · arxiv updated 2010/08/12
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.