Minimal partitions of a box into boxes (Q812789)

A box is a set of the form \(X=X_1\times\cdots\times X_d\), for some finite sets \(X_i\), \(i=1,\dots ,d\). \textit{N. Alon, T. Bohman, R. Holzman} and \textit{D. J. Kleitman} [Discrete Math. 257, 255--258 (2002; Zbl 1034.52021)] showed that any partition of \(X\) into nonempty sets of the form \(A_1\times\cdots \times A_d\), with \(A_i\subset X_i\) (\(A_i\neq X_i\)) must contain at least \(2^d\) members. In this paper the properties of such partitions with minimum possible number of parts are explored.