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

TL;DR

This paper introduces a new class of geometrically defined hypergraphs that relate to the longstanding Hadwiger Nelson problem, providing insights into their coloring properties and extending results to general normed spaces.

Contribution

It constructs a novel class of hypergraphs linked to the Hadwiger Nelson problem and generalizes the coloring results to arbitrary normed vector spaces.

Findings

Hypergraphs with arbitrarily large edge size linked to unit distance graph colorings

Proper colorings of these hypergraphs match those of the Euclidean unit distance graph

Partial generalizations to normed vector spaces

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.