On the counterexamples to Borsuk's conjecture by Kahn and Kalai

TL;DR

This paper revisits Kahn and Kalai's counterexamples to Borsuk's conjecture, correcting their derivation and providing a more precise formula to identify counterexamples in specific dimensions.

Contribution

It offers a detailed, formal correction to the derivation of counterexamples to Borsuk's conjecture, refining the dimension bounds for these counterexamples.

Findings

Corrected the derivation of the counterexample formula

Extended the validity range of counterexamples up to dimension 1560

Provided a more rigorous mathematical foundation for the counterexamples

Abstract

In the concluding remarks of their 1993 published and now famous paper, Jeff Kahn and Gil Kalai wrote in particular: "Our construction shows that Borsuk's conjecture is false for d = 1,325 and for every d > 2,014." But, as Bernulf Weiszbach remarked in his paper from 2000, a simple (few steps for an upper-class pocket calculator) computation indicates that that claim is not true for d=1325. William Kretschmer (Univ. of Texas) sent me his writeup on that paper by Kahn and Kalai, in particular pointing out that the derivation of the formula used in that computation disregarded a certain aspect, that way missed the chance to remove one final halving from that formula and to indeed provide a proof that a certain point set is a counterexample for dimension 1325 (and all higher dimensions up to 1560, too). This updated article takes a closer look at that derivation, gives an own, much more detailed and formal version of it that delivers the improved/corrected formula, and contains some further conclusions.