Minimal Partitions Of A Box Into Boxes

A box is a set of the form X = X 1×···×X d , for some finite sets X i , i = 1, . . .,d. Answering a question posed by Kearnes and Kiss [2], Alon, Bohman, Holzman and Kleitman proved [1] that any partition of X into nonempty sets of the form A 1×···×A d , with \( A_{i} \varsubsetneq X_{i} \), must contain at least 2d members. In this paper we explore properties of such partitions with minimum possible number of parts. In particular, we derive two characterizations of minimal partitions among all partitions of X into proper boxes. For instance, let P = P 1×···×P d be a fixed k-dimensional plane in X, that is P i = X i for exactly k different subscripts i, with |P i | = 1 otherwise. It is shown that \( {\user1{F}} \) is a minimal partition of X if and only if P intersects exactly 2k members of \( {\user1{F}} \), for every such P.

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Correspondence to Jarosław Grytczuk.

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Grytczuk, J., Kisielewicz, A.P. & Przesławski, K. Minimal Partitions Of A Box Into Boxes. Combinatorica 24, 605–614 (2004).

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Mathematics Subject Classification (2000):

  • 05A18
  • 52C22