Coloring Geometric Hypergraphs: A Survey
G\'abor Dam\'asdi, Bal\'azs Keszegh, J\'anos Pach, D\"om\"ot\"or P\'alv\"olgyi, G\'eza T\'oth

TL;DR
This survey reviews recent results on coloring geometric hypergraphs, focusing on their chromatic numbers and related covering problems in Euclidean spaces, highlighting the interplay between geometric configurations and combinatorial coloring questions.
Contribution
It compiles and discusses recent advances in understanding the chromatic properties of geometric hypergraphs, emphasizing the duality between covering and coloring problems in Euclidean geometry.
Findings
Chromatic number of geometric hypergraphs can often be bounded by small integers.
Certain geometric configurations allow 2-colorings that avoid monochromatic edges.
Large enough intersections in geometric hypergraphs tend to have chromatic number 2.
Abstract
The \emph{chromatic number} of a hypergraph is the smallest number of colors needed to color the vertices such that no edge of at least two vertices is monochromatic. Given a family of geometric objects that covers a subset of the Euclidean space, we can associate it with a hypergraph whose vertex set is and whose edges are those subsets for which there exists a point such that consists of precisely those elements of that contain . The question whether can be split into 2 coverings is equivalent to asking whether the chromatic number of the hypergraph is equal to 2. There are a number of competing notions of the chromatic number that lead to deep combinatorial questions already for abstract hypergraphs. In this paper, we concentrate on \emph{geometrically defined}…
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Taxonomy
TopicsAdvanced Graph Theory Research · Limits and Structures in Graph Theory · Computational Geometry and Mesh Generation
