Rainbow variations on a theme by Mantel: extremal problems for Gallai colouring templates
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929237314764800 |
|---|---|
| author | Falgas-Ravry, Victor Markström, Klas Räty, Eero |
| author_facet | Falgas-Ravry, Victor Markström, Klas Räty, Eero |
| contents | Let $\mathbf{G}:=(G_1, G_2, G_3)$ be a triple of graphs on the same vertex set $V$ of size $n$. A rainbow triangle in $\mathbf{G}$ is a triple of edges $(e_1, e_2, e_3)$ with $e_i\in G_i$ for each $i$ and $\{e_1, e_2, e_3\}$ forming a triangle in $V$. The triples $\mathbf{G}$ not containing rainbow triangles, also known as Gallai colouring templates, are a widely studied class of objects in extremal combinatorics.
In the present work, we fully determine the set of edge densities $(α_1, α_2, α_3)$ such that if $\vert E(G_i)\vert> α_i n^2$ for each $i$ and $n$ is sufficiently large, then $\mathbf{G}$ must contain a rainbow triangle. This resolves a problem raised by Aharoni, DeVos, de la Maza, Montejanos and Šámal, generalises several previous results on extremal Gallai colouring templates, and proves a recent conjecture of Frankl, Györi, He, Lv, Salia, Tompkins, Varga and Zhu. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_07180 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Rainbow variations on a theme by Mantel: extremal problems for Gallai colouring templates Falgas-Ravry, Victor Markström, Klas Räty, Eero Combinatorics Discrete Mathematics Let $\mathbf{G}:=(G_1, G_2, G_3)$ be a triple of graphs on the same vertex set $V$ of size $n$. A rainbow triangle in $\mathbf{G}$ is a triple of edges $(e_1, e_2, e_3)$ with $e_i\in G_i$ for each $i$ and $\{e_1, e_2, e_3\}$ forming a triangle in $V$. The triples $\mathbf{G}$ not containing rainbow triangles, also known as Gallai colouring templates, are a widely studied class of objects in extremal combinatorics. In the present work, we fully determine the set of edge densities $(α_1, α_2, α_3)$ such that if $\vert E(G_i)\vert> α_i n^2$ for each $i$ and $n$ is sufficiently large, then $\mathbf{G}$ must contain a rainbow triangle. This resolves a problem raised by Aharoni, DeVos, de la Maza, Montejanos and Šámal, generalises several previous results on extremal Gallai colouring templates, and proves a recent conjecture of Frankl, Györi, He, Lv, Salia, Tompkins, Varga and Zhu. |
| title | Rainbow variations on a theme by Mantel: extremal problems for Gallai colouring templates |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2212.07180 |