Partition problems in high dimensional boxes

TL;DR

This paper investigates the minimum number of sub-boxes needed to partition high-dimensional boxes, improving bounds from the trivial $3^d$ to approximately $2.93^d$, and explores related generalizations.

Contribution

It provides a new upper bound of about $2.93^d$ for partitioning odd-sized high-dimensional boxes, refining previous trivial bounds.

Findings

Approximately $2.93^d$ boxes suffice for partitioning

Improved bounds over the trivial $3^d$ construction

Exploration of natural generalizations of the problem

Abstract

Alon, Bohman, Holzman and Kleitman proved that any partition of a $d$-dimensional discrete box into proper sub-boxes must consist of at least $2^d$ sub-boxes. Recently, Leader, Mili\'{c}evi\'{c} and Tan considered the question of how many odd-sized proper boxes are needed to partition a $d$-dimensional box of odd size, and they asked whether the trivial construction consisting of $3^d$ boxes is best possible. We show that approximately $2.93^d$ boxes are enough, and consider some natural generalisations.