The basic problems in the classical theory of packings and coverings, the development of
which was strongly influenced by the geometry of numbers and by crystallography, are the …

A range family $\mathcal {R} $ is a family of subsets of $\mathbb {R}^ d $, like all halfplanes,
or all unit disks. Given a range family $\mathcal {R} $, we consider the $ m $-uniform range …

We prove that the intersection hypergraph of a family of n pseudo-disks with respect to
another family of pseudo-disks admits a proper coloring with four colors and a conflict-free …

The goal of this paper is to give a new, abstract approach to cover-decomposition and
polychromatic colorings using hypergraphs on ordered vertex sets. We introduce an abstract …

What is the minimum number of colors that always suffice to color every planar set of points
such that any disk that contains enough points contains two points of different colors? It is …

We prove that for every positive integer m there is a finite point set P in the plane such that
no matter how P is three-colored, there is always a disk containing exactly m points, all of the …

A system of sets forms an m-fold covering of a set X if every point of X belongs to at least m
of its members. A 1-fold covering is called a covering. The problem of splitting multiple …

We prove that for any planar convex body C there is a positive integer m with the property
that any finite point set P in the plane can be three-colored in such a way that no translate of …

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We prove that octants are cover-decomposable; ie, any 12-fold covering of any subset of the
space with a finite number of translates of a given octant can be decomposed into two …