Nearly neighbourly families of standard boxes

TL;DR

This paper proves that nearly neighbourly families of standard boxes in three-dimensional space have a maximum of 12 elements, classifies all such maximal families, and discusses properties like incompressibility.

Contribution

It establishes an upper bound of 12 for nearly neighbourly families of standard boxes in 3D and provides a complete classification of maximal families.

Findings

Maximum of 12 elements in nearly neighbourly families

Complete classification of families with exactly 12 elements

Discussion of incompressible and compressible families

Abstract

It is demonstrated that each nearly neighbourly family of standard boxes in $\mathbb{R}^3$ has at most 12 elements. A combinatorial classification of all such families that have exactly 12 elements is given. All families satisfying an extra property called incompressibility are described. Compressible families are discussed briefly.