Neighborhood Variants of the KKM Lemma, Lebesgue Covering Theorem, and Sperner's Lemma on the Cube

TL;DR

This paper introduces neighborhood variants of classical topological lemmas like KKM, Lebesgue, and Sperner's lemma on the cube, providing discretized results that guarantee multiple colors within small neighborhoods under certain coloring conditions.

Contribution

It establishes new neighborhood versions of key topological lemmas on the cube and derives a discretized Sperner's lemma variant with quantitative color diversity guarantees.

Findings

Existence of an $ ext{l}_ ext{infty}$ $ ext{varepsilon}$-ball with many colors

Quantitative bounds on the number of colors in neighborhoods

Neighborhood variants of classical lemmas on the cube

Abstract

We establish a "neighborhood" variant of the cubical KKM lemma and the Lebesgue covering theorem and deduce a discretized version which is a "neighborhood" variant of Sperner's lemma on the cube. The main result is the following: for any coloring of the unit $d$-cube $[0,1]^d$ in which points on opposite faces must be given different colors, and for any $\varepsilon>0$, there is an $\ell_\infty$ $\varepsilon$-ball which contains points of at least $(1+\frac{\varepsilon}{1+\varepsilon})^d$ different colors, (so in particular, at least $(1+\frac{2}{3}\varepsilon)^d$ different colors for all sensible $\varepsilon\in(0,\frac12]$).