The size of maximal systems of brick islands

For integers $m_1,\mathrm{.}\mathrm{.}\mathrm{.},m_d>0$ and a cuboid $M=[0,m_1]\times … \times [0,m_d]\subset $\mathbb{R}$^d$,abrickof$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 $\sum_{i=1}^d m_i - (d-1)$\mathrm{.}Also,inacube$C=[0,m]^d$ we define the corresponding notion of a system of cubic islands, and prove bounds on the sizes of maximal systems of cubic islands.