On the Borsuk number of four-dimensional sets

TL;DR

This paper improves the upper bound on the number of smaller diameter subsets needed to partition four-dimensional sets, refining the previous estimate from nine to eight, thus advancing understanding of Borsuk's conjecture in four dimensions.

Contribution

The paper provides a tighter upper bound for the Borsuk number of four-dimensional sets, reducing it from nine to eight.

Findings

Improved the upper bound for four-dimensional sets from nine to eight.

Refined the understanding of Borsuk's conjecture in four dimensions.

Contributes to the ongoing investigation of Borsuk numbers in higher dimensions.

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.