Irreducible subcube partitions

TL;DR

This paper investigates the properties and extremal characteristics of tight irreducible subcube partitions in Boolean and related spaces, exploring minimal and maximal configurations, and proposing conjectures based on constructions and experiments.

Contribution

It introduces new extremal properties of tight irreducible subcube partitions and explores their existence, minimality, and maximality across various settings, supported by constructions and computational experiments.

Findings

Identified bounds for minimal size and weight of tight irreducible partitions.

Constructed examples and conducted experiments leading to conjectures on extremal values.

Analyzed partitions of different spaces, including Boolean cubes and affine subspaces.

Abstract

A \emph{subcube partition} is a partition of the Boolean cube $\{0,1\}^n$ into subcubes. A subcube partition is irreducible if the only sub-partitions whose union is a subcube are singletons and the entire partition. A subcube partition is tight if it "mentions" all coordinates. We study extremal properties of tight irreducible subcube partitions: minimal size, minimal weight, maximal number of points, maximal size, and maximal minimum dimension. We also consider the existence of homogeneous tight irreducible subcube partitions, in which all subcubes have the same dimensions. We additionally study subcube partitions of $\{0,\dots,q-1\}^n$, and partitions of $\mathbb{F}_2^n$ into affine subspaces, in both cases focusing on the minimal size. Our constructions and computer experiments lead to several conjectures on the extremal values of the aforementioned properties.