Colorful Borsuk--Ulam theorems and applications

TL;DR

This paper introduces colorful generalizations of fundamental topological theorems like Borsuk--Ulam, ham sandwich, and Brouwer's fixed point, with applications to measure separation, sphere coverings, and symmetry.

Contribution

It presents novel colorful versions of key theorems, extending classical results and providing new tools for topological and combinatorial applications.

Findings

Generalized Borsuk--Ulam theorem with colorful variants

Colorful ham sandwich theorem and its strengthening

Alternative between Radon-type and KKM-type results

Abstract

We prove a colorful generalization of the Borsuk--Ulam theorem and derive colorful consequences from it, such as a colorful generalization of the ham sandwich theorem. Even in the uncolored case this specializes to a strengthening of the ham sandwich theorem, which given an additional condition, contains a result of B\'{a}r\'{a}ny, Hubard, and Jer\'{o}nimo on well-separated measures as a special case. We prove a colorful generalization of Fan's antipodal sphere covering theorem, we derive a short proof of Gale's colorful KKM theorem, and we prove a colorful generalization of Brouwer's fixed point theorem. Our results also provide an alternative between Radon-type intersection results and KKM-type covering results. Finally, we prove colorful Borsuk--Ulam theorems for higher symmetry.