# Coloring hypergraphs defined by stabbed pseudo-disks and ABAB-free hypergraphs

## TL;DR

This paper proves that hypergraphs defined by stabbed pseudo-disks and ABAB-free hypergraphs are always 3-colorable, resolving a specific coloring problem for planar point sets and their topological generalizations.

## Contribution

It establishes that ABAB-free hypergraphs, which characterize stabbed pseudo-disk hypergraphs, are always 3-chromatic, providing a combinatorial proof for a geometric coloring problem.

## Key findings

- 3 colors suffice for stabbing pseudo-disk hypergraphs
- ABAB-free hypergraphs are equivalent to stabbed pseudo-disk hypergraphs
- The proof is purely combinatorial and tight for the given conditions

## 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 known that the answer to this question is either three or four. We show that three colors always suffice if the condition must be satisfied only by disks that contain a fixed point. Our result also holds, and is even tight, when instead of disks we consider their topological generalization, namely pseudo-disks, with a non-empty intersection. Our solution uses the equivalence that a hypergraph can be realized by stabbed pseudo-disks if and only if it is ABAB-free. These hypergraphs are defined in a purely abstract, combinatorial way and our proof that they are 3-chromatic is also combinatorial.