Brick partition problems in three dimensions

TL;DR

This paper investigates minimal brick partitions in three dimensions that ensure each axis line or plane intersects a specified number of bricks, providing near-optimal constructions and exact solutions for certain intersection conditions.

Contribution

It nearly determines the minimal size of 3D brick partitions with k-piercing properties and exactly solves the minimal size for partitions intersected by planes in three dimensions.

Findings

Constructed a 3D brick partition with 12k-15 parts for k-piercing.

Established the minimal size s(3, k) for plane intersections in 3D.

Resolved the minimal partition size problem for 3D with plane intersection constraints.

Abstract

A $d$-dimensional brick is a set $I_1\times \cdots \times I_d$ where each $I_i$ is an interval. Given a brick $B$, a brick partition of $B$ is a partition of $B$ into bricks. A brick partition $\mathcal{P}_d$ of a $d$-dimensional brick is $k$-piercing if every axis-parallel line intersects at least $k$ bricks in $\mathcal{P}_d$. Bucic et al. explicitly asked the minimum size $p(d, k)$ of a $k$-piercing brick partition of a $d$-dimensional brick. The answer is known to be $4(k-1)$ when $d=2$. Our first result almost determines $p(3, k)$. Namely, we construct a $k$-piercing brick partition of a $3$-dimensional brick with $12k-15$ parts, which is off by only $1$ from the known lower bound. As a generalization of the above question, we also seek the minimum size $s(d, k)$ of a brick partition $\mathcal{P}_d$ of a $d$-dimensional brick where each axis-parallel plane intersects at least $k$ bricks in $\mathcal{P}_d$. We resolve the question in the $3$-dimensional case by determining $s(3, k)$ for all $k$.