Clouds in Higher Dimensions

TL;DR

This paper extends the concept of clouds from two-dimensional space to higher dimensions, establishing conditions for coverage and demonstrating that countably many clouds can cover any higher-dimensional space.

Contribution

It generalizes the definition of clouds to -dimensional spaces and proves coverage equivalences and countability results in higher dimensions.

Findings

k clouds cover -dimensional space iff they cover -dimensional space

Countably many clouds can cover -dimensional space

Coverage properties are preserved across dimensions

Abstract

Following some work done by Komjath and Schmerl, we extend the definition of a cloud to $\mathbb{R}^N$ for $N \geq 2$ and show that $k$ clouds cover $\mathbb{R}^2$ if and only if $k$ clouds cover $\mathbb{R}^N$. We also show that countably many clouds cover $\mathbb{R}^N$.