The structure of rectangle families dividing the plane into maximum number of atoms

Let .s4 = {Al, ... 'An} be a family of sets. The elements X, y E u~=l Ai are called equivalent if for every i, 1:::;; i :::;; n, x E Ai if and only if y E Ai. The equivalence classes are called the atoms of the family d. Rado asked in [ 4] the following question: what is the maximum number f(n, d) of atoms, where the maximum is taken over families of n boxes in the d-dimensional Euclidean space. A box is a parallelepiped with sides parallel to the coordinate axes. A family of n boxes is extremal if it defines f(n, d) atoms. Rado showed that f(n, 1) = 2n -1. The authors of the present paper proved that f(n, 2) = 2n -6n +7 if n ~2, determined f(n, 3) asymptotically and gave upper and lower bounds for f(n, d) (see [3]). The present paper is devoted to the two-dimensional extremal families of boxes which we call box diagrams. Our main result, Theorem 3.1, is the characterization of box diagrams. It turns out that all box diagrams can be obtained by a slight modification (peripheral lifting) from a basic type: the caterpillar construction given in Section 2. Box diagrams defined by the caterpillar construction for n = 3 and n = 4 are shown in Figs. 6 and 9 in Section 2. We note that this characterization describes the structure of box diagrams completely. It is remarkable that one-dimensional extremal families have no structural characterization. As proved in [3], these are interval families with connected overlap graphs for which only a non-structural characterization is known (cf. [2]). We show two consequences of the main result. The first one concerns the enumeration of box diagrams: apart from axial symmetries, there are