Coloring Geometric Hypergraphs: A Survey

TL;DR

This survey analyzes how geometric hypergraphs arising from translates, homothets, and other geometric objects can be colored. It surveys a spectrum of colorings (proper, polychromatic, conflict-free) and links colorability to covering decomposability, using tools like self-coverability, Delaunay graphs, and ABA-free frameworks. Major results identify when the big chromatic number is bounded (often by 2, 3, or 4) for broad geometric families, while outlining tight bounds and open problems for axis-aligned shapes, disks, and intersection hypergraphs. The work highlights both universal techniques and shape-specific phenomena, offering a roadmap for understanding decomposability of coverings and colorings in geometric settings.

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 $\mathcal{F}$ that covers a subset $S$ of the Euclidean space, we can associate it with a hypergraph whose vertex set is $\mathcal F$ and whose edges are those subsets ${\mathcal{F}'}\subset \mathcal F$ for which there exists a point $p\in S$ such that ${\mathcal F}'$ consists of precisely those elements of $\mathcal{F}$ that contain $p$. The question whether $\mathcal F$ 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} (in short, \emph{geometric}) hypergraphs, and survey many recent coloring results related to them. In particular, we study and survey the following problem, dual to the above covering question. Given a set of points $S$ in the Euclidean space and a family $\mathcal{F}$ of geometric objects of a fixed type, define a hypergraph ${\mathcal H}_m$ on the point set $S$, whose edges are the subsets of $S$ that can be obtained as the intersection of $S$ with a member of $\mathcal F$ and have at least $m$ elements. Is it true that if $m$ is large enough, then the chromatic number of ${\mathcal H}_m$ is equal to 2?

Figures (8)

• Figure 1: An example of the extension operation.
• Figure 2: The hypergraph $\mathcal{H}(3,3)$ with a 2-coloring. In this case there is a red edge whose vertices are all red.
• Figure 3: Octants give a richer family than homothetic copies of a triangle, because every homothet of the triangle depicted on the shaded plane can be obtained as the intersection of an octant with the plane.
• Figure 4: A shift-chain of $13$ triples, each of which corresponds to a row. For any $2$-coloring of the $9$ vertices, one of the triples is monochromatic.
• Figure 5: The inclusion hierarchy of hypergraph families defined by disks and related objects and the best known results about their respective coloring parameters.

Theorems & Definitions (50)

• Conjecture 2.1
• Lemma 2.2: Damásdi-Pálvölgyi DP22
• proof
• Corollary 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.7
• Theorem 3.1: Pach pach1986
• Theorem 3.2: Tardos-Tóth TT07
• Theorem 3.3: Pálvölgyi-Tóth PT10