On colorings of hypergraphs embeddable in $\mathbb{R}^d$
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908872239742976 |
|---|---|
| author | Lee, Seunghun Nevo, Eran |
| author_facet | Lee, Seunghun Nevo, Eran |
| contents | 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 $1\leq s \leq d$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_14195 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On colorings of hypergraphs embeddable in $\mathbb{R}^d$ Lee, Seunghun Nevo, Eran Combinatorics 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 $1\leq s \leq d$. |
| title | On colorings of hypergraphs embeddable in $\mathbb{R}^d$ |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2307.14195 |