On Partitions of a Rectangle into Rectangles with Restricted Number of Cross Sections

Consider partitions of a rectangle into rectangles with restricted number of cross sections. The problem has an information theoretic flavor (in the spirit of [3]): richness of two dimensional pictures under 1–dimensional restrictions of from local information to global information. We present two approaches to upper bound the cardinality of such partitions: a combinatorial recursion and harmonic analysis. 1 The Problem Given a partition of a rectangle into rectangles, such that any line parallel to a side of the initial rectangle intersects at most n rectangles of the partition. The task is to estimate the maximal number of rectangles of the partition. We make now our concept of partition precise and consider a more general problem. Definition 1. We say that a family of rectangles S gives a partition of the rectangle F , if ⋃ A∈S A = F (1.1) and for ∀A,B ∈ S, A = B the intersection of A and B contains at most boundary points. Definition 2. We say that the partition SF has the property (m,n), if any line parallel to the base of the rectangle F intersects at most m rectangles of the partition, and perpendicular to the base — at most n rectangles. The problem is to find f(m,n) = sup card(SF ) = sup |SF |, (1.2) where the supremum is taken over all partitions SF with property (m,n). It is evident that f(m,n) = f(n,m). In case m = n one readily sees that f(n, n) ≥ 2 f(n− 1, n− 1) + 2 ≥ 2 for n ≥ 2. (1.3) It seems natural to conjecture that the first inequality is actually an equality. By successively improving our methods we derive here a decreasing sequence of upper bounds on f(m,n) and finally come close to the conjectured bound. Recently we learnt that a paper on such problems appeared already in 1993: D.J. Kleitman “Partitioning a rectangle into many sub-rectangles so that a line can meet only a few”, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Volume 9, 95-107, 1993. R. Ahlswede et al. (Eds.): Information Transfer and Combinatorics, LNCS 4123, pp. 941–954, 2006. c © Springer-Verlag Berlin Heidelberg 2006 942 R. Ahlswede and A.A. Yudin 2 A Crude Estimate Let us look at