Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Coloring equilateral triangles

Jindrich Zapletal

TL;DR

This work studies chromatic properties of 3-ary algebraic hypergraphs defined by triangle configurations in Euclidean spaces. It combines real algebraic geometry with a sigma-closed, definable coloring poset within choiceless Solovay-type models to bound colorings across dimensions. Under the assumption that ZFC plus an inaccessible cardinal is consistent, it shows that the equilateral-triangle hypergraph in any dimension has countable chromatic number, while the isosceles-triangle hypergraph on the plane does not, and proves DC in the resulting model. The results illustrate independence phenomena for geometric hypergraphs in choiceless settings and suggest avenues for extending the analysis to broader algebraic hypergraphs and higher dimensions.

Abstract

It is consistent relative to an inaccessible cardinal that ZF+DC holds, the hypergraph of equilateral triangles on a given Euclidean space has countable chromatic number, while the hypergraph of isosceles triangles in the plane does not.

Coloring equilateral triangles

TL;DR

This work studies chromatic properties of 3-ary algebraic hypergraphs defined by triangle configurations in Euclidean spaces. It combines real algebraic geometry with a sigma-closed, definable coloring poset within choiceless Solovay-type models to bound colorings across dimensions. Under the assumption that ZFC plus an inaccessible cardinal is consistent, it shows that the equilateral-triangle hypergraph in any dimension has countable chromatic number, while the isosceles-triangle hypergraph on the plane does not, and proves DC in the resulting model. The results illustrate independence phenomena for geometric hypergraphs in choiceless settings and suggest avenues for extending the analysis to broader algebraic hypergraphs and higher dimensions.

Abstract

It is consistent relative to an inaccessible cardinal that ZF+DC holds, the hypergraph of equilateral triangles on a given Euclidean space has countable chromatic number, while the hypergraph of isosceles triangles in the plane does not.

Paper Structure

This paper contains 4 sections, 13 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. Algebraic geometry
  3. The coloring poset
  4. The independence result

Key Result

Theorem 1.1

If the theory ZFC+there is an inaccessible cardinal is consistent, then so is ZF+DC+for every number $d\geq 1$, the hypergraph of equilateral triangles in dimension $d$ has countable chromatic number+the hypergraph of isosceles triangles in $\mathbb{R}^2$ does not have countable chromatic number.

Theorems & Definitions (43)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.3
  • Example 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • proof
  • Proposition 2.7
  • proof
  • ...and 33 more