Multilabeled versions of Sperner's and Fan's lemmas and applications

TL;DR

This paper introduces a versatile technique for multilabeled Sperner and Fan's lemmas, enabling new applications in fair division, cake-cutting, and graph coloring, expanding the scope of classical combinatorial lemmas.

Contribution

It develops a general method for multilabeled versions of Sperner's and Fan's lemmas, with novel applications in cake-cutting and graph coloring.

Findings

A new cake-cutting theorem with independent control of players and pieces

Multilabeled Fan's lemma versions applicable to fair division and graph coloring

Enhanced combinatorial tools for multilabeled topological lemmas

Abstract

We propose a general technique related to the polytopal Sperner lemma for proving old and new multilabeled versions of Sperner's lemma. A notable application of this technique yields a cake-cutting theorem where the number of players and the number of pieces can be independently chosen. We also prove multilabeled versions of Fan's lemma, a combinatorial analogue of the Borsuk-Ulam theorem, and exhibit applications to fair division and graph coloring.