The size of maximal systems of brick islands

TL;DR

This paper determines the minimum size of maximal systems of brick islands within a cuboid and explores bounds for cubic islands in a cube, extending previous work in combinatorial geometry.

Contribution

It establishes the exact minimum size of maximal brick island systems and provides bounds for cubic island systems, advancing understanding in geometric combinatorics.

Findings

Minimum size of maximal brick island systems is $ extstyle\sum_{i=1}^d m_i - (d-1)$.

Derived bounds for maximal systems of cubic islands in a cube.

Extended previous results by Lengvárszky to higher dimensions.

Abstract

For integers $m_1,...,m_d>0$ and a cuboid $M=[0,m_1]\times ... \times [0,m_d]\subset \mathbb{R}^d$, a brick of $M$ is a closed cuboid whose vertices have integer coordinates. A set $H$ of bricks in $M$ is a system of brick islands if for each pair of bricks in $H$ one contains the other or they are disjoint. Such a system is maximal if it cannot be extended to a larger system of brick islands. Extending the work of Lengv\'{a}rszky, we show that the minimum size of a maximal system of brick islands in $M$ is $\sum_{i=1}^d m_i - (d-1)$. Also, in a cube $C=[0,m]^d$ we define the corresponding notion of a system of cubic islands, and prove bounds on the sizes of maximal systems of cubic islands.