A two-page disproof of the Borsuk partition conjecture

TL;DR

This paper presents the simplest known disproof of the Borsuk conjecture using combinatorics and algebra, making the complex proof accessible for students.

Contribution

It provides an accessible, two-page disproof of the Borsuk conjecture, highlighting a novel application of combinatorics and algebra to geometry.

Findings

Disproof of the Borsuk conjecture for all dimensions

Simplified, two-page proof accessible to students

Application of combinatorics and algebra to geometric problems

Abstract

It is presented the simplest known disproof of the Borsuk conjecture stating that if a bounded subset of n-dimensional Euclidean space contains more than n points, then the subset can be partitioned into n+1 nonempty parts of smaller diameter. The argument is due to N. Alon and is a remarkable application of combinatorics and algebra to geometry. This note is purely expository and is accessible for students.