The size of maximal systems of brick islands

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 Lengvárszky, 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.

12 pages