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Three-chromatic geometric hypergraphs

Gábor Damásdi, Dömötör Pálvölgyi

TL;DR

It is proved 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 such that there is no translate of C containing at least m points of P , all of the same color.

Abstract

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 such that there is no translate of C containing at least m points of P, all of the same color. As a part of the proof, we show a strengthening of the Erdős-Sands-Sauer-Woodrow conjecture. Surprisingly, the proof also relies on the two dimensional case of the Illumination conjecture.

Three-chromatic geometric hypergraphs

TL;DR

It is proved 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 such that there is no translate of C containing at least m points of P , all of the same color.

Abstract

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 such that there is no translate of C containing at least m points of P, all of the same color. As a part of the proof, we show a strengthening of the Erdős-Sands-Sauer-Woodrow conjecture. Surprisingly, the proof also relies on the two dimensional case of the Illumination conjecture.
Paper Structure (10 sections, 16 theorems, 10 equations, 5 figures)

This paper contains 10 sections, 16 theorems, 10 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Tools
  3. Union Lemma
  4. Erdős-Sands-Sauer-Woodrow conjecture
  5. Hadwiger's Illumination conjecture and pseudolines
  6. Proof of Theorem \ref{['thm:general_three_col']}
  7. Quasi-orders on planar point sets
  8. Proof of Theorem \ref{['thm:general_three_col']}
  9. Proof of Theorem \ref{['thm:multi_essw_new']}
  10. Open questions

Key Result

Theorem 1.1

For any planar convex body $C$ there is a positive integer $m=m(C)$ such that any finite point set $P$ in the plane can be three-colored in a way that there is no translate of $C$ containing at least $m$ points of $P$, all of the same color.

Figures (5)

  • Figure 1: Extremal tangents at a boundary point (on the left) and parallel tangents on two intersecting translates (on the right).
  • Figure 2: Basic properties of cones.
  • Figure 3: Quasi-order on a point set.
  • Figure 4: Selecting the cones (on the left) and Property (A) (on the right).
  • Figure 5: The outcome of the partition process for $k=3$ assuming that the size of $T_3$ and $T_{1,3}$ is less than $1/\delta$.

Theorems & Definitions (27)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.4: Damásdi, Pálvölgyi fourchromatic
  • Theorem 1.5: Pach, Pálvölgyi unsplittable
  • Lemma 2.1: Union Lemma
  • proof
  • Theorem 2.2: Keszegh-Pálvölgyi abafree
  • Corollary 2.3
  • Theorem 2.4: Bousquet, Lochet, Thomassé esswproof
  • Theorem 2.5
  • ...and 17 more