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 …

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 …

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 …

We consider a search problem on trees in which the goal is to find an adversarially placed
treasure, while relying on local, partial information. Specifically, each node in the tree holds …

In this paper we study the minimal size of edges in hypergraph families which guarantees
the existence of a polychromatic coloring, that is, a $ k $-coloring of a vertex set such that …

In this note we improve our upper bound given earlier by showing that every 9-fold covering
of a point set in the space by finitely many translates of an octant decomposes into two …

A polychromatic k-coloring of a hypergraph assigns to each vertex one of k colors in such a
way that every hyperedge contains all the colors. A range capturing hypergraph is an m …

We give new positive results on the long-standing open problem of geometric covering
decomposition for homothetic polygons. In particular, we prove that for any positive integer k …