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A New Class of Geometrically Defined Hypergraphs Arising from the Hadwiger Nelson Problem

Sean Fiscus, Eric Myzelev, Hongyi Zhang

Abstract

There is a famous problem in geometric graph theory to find the chromatic number of the unit distance graph on Euclidean space; it remains unsolved. A theorem of Erdos and De-Bruijn simplifies this problem to finding the maximum chromatic number of a finite unit distance graph. Via a construction built on sequential finite graphs obtained from a generalization of this theorem, we have found a class of geometrically defined hypergraphs of arbitrarily large edge cardinality, whose proper colorings exactly coincide with the proper colorings of the unit distance graph on $\mathbb R^d$. We also provide partial generalizations of this result to arbitrary real normed vector spaces.

A New Class of Geometrically Defined Hypergraphs Arising from the Hadwiger Nelson Problem

Abstract

There is a famous problem in geometric graph theory to find the chromatic number of the unit distance graph on Euclidean space; it remains unsolved. A theorem of Erdos and De-Bruijn simplifies this problem to finding the maximum chromatic number of a finite unit distance graph. Via a construction built on sequential finite graphs obtained from a generalization of this theorem, we have found a class of geometrically defined hypergraphs of arbitrarily large edge cardinality, whose proper colorings exactly coincide with the proper colorings of the unit distance graph on . We also provide partial generalizations of this result to arbitrary real normed vector spaces.

Paper Structure

This paper contains 5 sections, 10 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. De Bruijn-Erdős for hypergraphs with finite edges
  3. Hypergraphs Equivalent to the Unit Distance Graph
  4. Generalizing to Non-Euclidean Norms on $\mathbb{R}^d$
  5. Explicit Construction of Hypergraph

Key Result

Theorem 2.1

Suppose that $m\in \mathbb{Z}^+$, $\mathcal{H}=(V,E)$ is a hypergraph with $2\leq |e|<\infty$ for every $e\in E$, and $\chi(\mathcal{H}|_F)\leq m$ for every finite subset $F$ of $V$. Then $\chi(\mathcal{H})\leq m$.

Theorems & Definitions (20)

  • Theorem 2.1: D-E
  • Remark 2.1.1
  • proof
  • Definition 3.1
  • Theorem 3.1
  • proof
  • Corollary 3.1.1
  • proof
  • Lemma 4.1
  • Lemma 4.2
  • ...and 10 more