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On colorings of hypergraphs embeddable in $\mathbb{R}^d$

Seunghun Lee, Eran Nevo

Abstract

The (weak) chromatic number of a hypergraph $H$, denoted by $χ(H)$, is the smallest number of colors required to color the vertices of $H$ so that no hyperedge of $H$ is monochromatic. For every $2\le k\le d+1$, denote by $χ_L(k,d)$ (resp. $χ_{PL}(k,d)$) the supremum $\sup_H χ(H)$ where $H$ runs over all finite $k$-uniform hypergraphs such that $H$ forms the collection of maximal faces of a simplicial complex that is linearly (resp. PL) embeddable in $\mathbb{R}^d$. Following the program by Heise, Panagiotou, Pikhurko and Taraz, we improve their results as follows: For $d \geq 3$, we show that A. $χ_L(k,d)=\infty$ for all $2\le k\le d$, B. $χ_{PL}(d+1,d)=\infty$ and C. $χ_L(d+1,d)\ge 3$ for all odd $d\ge 3$. As an application, we extend the results by Lutz and Møller on the weak chromatic number of the $s$-dimensional faces in the triangulations of a fixed triangulable $d$-manifold $M$: D. $χ_s(M)=\infty$ for $1\leq s \leq d$.

On colorings of hypergraphs embeddable in $\mathbb{R}^d$

Abstract

The (weak) chromatic number of a hypergraph , denoted by , is the smallest number of colors required to color the vertices of so that no hyperedge of is monochromatic. For every , denote by (resp. ) the supremum where runs over all finite -uniform hypergraphs such that forms the collection of maximal faces of a simplicial complex that is linearly (resp. PL) embeddable in . Following the program by Heise, Panagiotou, Pikhurko and Taraz, we improve their results as follows: For , we show that A. for all , B. and C. for all odd . As an application, we extend the results by Lutz and Møller on the weak chromatic number of the -dimensional faces in the triangulations of a fixed triangulable -manifold : D. for .
Paper Structure (9 sections, 10 theorems, 19 equations, 2 figures, 1 table)

This paper contains 9 sections, 10 theorems, 19 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. XL(k,d) is unbounded when 2<=k<=d and d>=3
  3. Combinatorial criterion for embeddability on the moment curve
  4. Proof of Theorem A
  5. XPL(d+1,d) is unbounded when d>=3
  6. XL(d+1,d)>=3 for odd d>=3
  7. Construction of Ld
  8. Non-2-colorability of Ld
  9. Embeddability of Ld

Key Result

Theorem 1.1

For every nonnegative integer $d$, if a simplicial complex $\mathsf{K}$ is PL embeddable in $\mathbb{R}^d$ then there exists a (PL) triangulation $\mathsf{T}$ of the $d$-simplex containing $\mathsf{K}$ as a subcomplex in its interior such that the boundary of $\mathsf{T}$ is combinatorially isomorph

Figures (2)

  • Figure 1: Illustration of the construction in the proof of Lemma \ref{['lemma_linear_PL-embed']} when $d=3$. Here, $f_1=\{v_1, v_2, v_3, v_4\}$ and $f_2=\{v_4, v_5, v_6, v_7\}$. The tetrahedrons $\sigma_{f_i}$ are indicated by bold edges, and their respective facets $\tau_{f_i}$ are indicated by light blue.
  • Figure 2: Illustration of the geometric embedding of $L_3$. A small circle around each end vertex of $s_i$ and $s_i'$ describes a small neighborhood where the other vertices are introduced. The blue lines indicate a thin tetrahedron in the realization of $M_1*M_2$. The other notations are introduced in the proof below.

Theorems & Definitions (22)

  • Theorem 1.1: adiprasito-patakova2024higherdimensionalversionfarystheoremPL-extension-karim-geva
  • Lemma 2.1
  • proof
  • Example 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4: ABAB_stabbed_pseudodisk
  • Proposition 2.5
  • proof : Proof of Theorem A
  • proof : Proof of Proposition \ref{['prop_ABABA-free']}
  • ...and 12 more