Coloring Geometric Hypergraphs: A Survey

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?