Summary: On Partitions of Discrete Boxes
Daniel J. Kleitman§
February 22, 2002
We prove that any partition of an n-dimensional discrete box into
nontrivial sub-boxes must consist of at least 2n
sub-boxes, and consider
some extensions of this theorem.
1 The theorem
A set of the form
A = A1 × A2 × · · · × An,
where A1, A2, . . . , An are finite sets with |Ai| 2, will be called here an
n-dimensional discrete box. A set of the form B = B1 × B2 × · · · × Bn,
where Bi Ai, i = 1, . . . , n, is a sub-box of A. Such a set B is said to be
nontrivial if = Bi = Ai for every i.
The following theorem answers a question posed by Kearnes and Kiss [1,