 
Summary: On Partitions of Discrete Boxes
Noga Alon
Tom Bohman
Ron Holzman
Daniel J. Kleitman§
February 22, 2002
Abstract
We prove that any partition of an ndimensional discrete box into
nontrivial subboxes must consist of at least 2n
subboxes, 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
ndimensional discrete box. A set of the form B = B1 × B2 × · · · × Bn,
where Bi Ai, i = 1, . . . , n, is a subbox 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,
Problem 5.5].
