Some old and new problems in combinatorial geometry I: Around Borsuk's problem

TL;DR

This paper discusses classical and recent open problems in combinatorial geometry related to Borsuk's problem, including historical context, solutions, and new questions in the field.

Contribution

It reviews existing solutions and introduces new questions concerning Borsuk's problem in combinatorial geometry.

Findings

Kahn and Kalai provided a negative answer to Borsuk's problem in 1993.

The paper highlights unresolved questions and directions for future research.

Historical overview of Borsuk's problem and related theorems.

Abstract

Borsuk asked in 1933 if every set of diameter 1 in $R^d$ can be covered by $d+1$ sets of smaller diameter. In 1993, a negative solution, based on a theorem by Frankl and Wilson, was given by Kahn and Kalai. In this paper I will present questions related to Borsuk's problem.