A topological proof of Ky Fan's covering lemma

TL;DR

This paper provides a combinatorial proof of Ky Fan's covering lemma using the Borsuk-Ulam theorem and extends the approach to multiple linear orders.

Contribution

It introduces a new combinatorial proof technique for Ky Fan's lemma and generalizes it to multiple linear orders.

Findings

Proof based on Borsuk-Ulam theorem

Extension to several linear orders

New combinatorial perspective

Abstract

We give an intuitive combinatorial proof of Ky Fan's covering lemma based on the Borsuk-Ulam theorem. We then show how this approach can be generalized to Ky Fan's covering lemma for several linear orders.