On partitions of discrete boxes (Q1850011)

A set of the form \(A=A_1 \times A_2 \times \dots \times A_n\), where \(A_1, A_2, \dots, A_n\) are finite sets with \(| A_i| \geq 2\), is called an \(n\)-dimensional discrete box. A set of the form \(B=B_1 \times B_2 \times \dots \times B_n\), where \(B_i \subseteq A_i\), \(i=1, \dots ,n\), is a sub-box of \(A\). Such a set \(B\) is said to be nontrivial if \(\emptyset \not= B_i \not= A_i\) for every \(i\). It is proved that any partition of an \(n\)-dimensional discrete box into nontrivial sub-boxes must consist of at least \(2^n\) sub-boxes. Furthermore, some extensions of this result are considered.