Brick partition problems in three dimensions

Abstract

A d-dimensional brick is a set I1×⋯×Id where each Ii is an interval. Given a brick B, a brick partition ofB is a partition of B into bricks. A brick partition Pd of a d-dimensional brick is k-piercing if every axis–parallel line intersects at least k bricks in Pd. Bucic et al. (2019) 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 Pd of a d-dimensional brick where each axis–parallel plane intersects at least k bricks in Pd. We resolve the question in the 3-dimensional case by determining s(3,k) for all k.