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Rainbow variations on a theme by Mantel: extremal problems for Gallai colouring templates

Victor Falgas-Ravry, Klas Markström, Eero Räty

TL;DR

This paper resolves the problem of characterising forcing triples $(\alpha_1,\alpha_2,\alpha_3)$ that guarantee a rainbow triangle in Gallai 3-colouring templates on $n$ vertices. The authors develop a two-regime, density-geometry framework using canonical representations $\alpha_1=x^2+y^2$, $\alpha_2=x^2+(1-x-y)^2$ (or $\alpha_3=1-x^2$) and $\alpha_3=2-\alpha_1-2\sqrt{\alpha_2}+\alpha_2$ to partition the parameter space into the $\mathbf{F}$-extremal and $\mathbf{H}$-extremal regions. They prove that in the $\mathbf{F}$-region, high sums of two colour-edge counts force a rainbow triangle, while the $\mathbf{F}$ templates show sharpness; in the $\mathbf{H}$-region, a suitable potential function $g(\mathbf{G})$ guarantees a rainbow triangle, with $\mathbf{H}$ templates attaining the lower bounds. Collectively, this fully determines the forcing set of triples and confirms a conjecture on the asymptotic extremal product of edge densities, strengthening Turán-type results for Gallai colourings and settling several prior conjectures. The results generalize earlier work of Keevash et al. and Aharoni et al., and open avenues for stability analyses and extensions to larger cliques and other graphs.

Abstract

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.

Rainbow variations on a theme by Mantel: extremal problems for Gallai colouring templates

TL;DR

This paper resolves the problem of characterising forcing triples that guarantee a rainbow triangle in Gallai 3-colouring templates on vertices. The authors develop a two-regime, density-geometry framework using canonical representations , (or ) and to partition the parameter space into the -extremal and -extremal regions. They prove that in the -region, high sums of two colour-edge counts force a rainbow triangle, while the templates show sharpness; in the -region, a suitable potential function guarantees a rainbow triangle, with templates attaining the lower bounds. Collectively, this fully determines the forcing set of triples and confirms a conjecture on the asymptotic extremal product of edge densities, strengthening Turán-type results for Gallai colourings and settling several prior conjectures. The results generalize earlier work of Keevash et al. and Aharoni et al., and open avenues for stability analyses and extensions to larger cliques and other graphs.

Abstract

Let be a triple of graphs on the same vertex set of size . A rainbow triangle in is a triple of edges with for each and forming a triangle in . The triples 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 such that if for each and is sufficiently large, then 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.
Paper Structure (11 sections, 24 theorems, 71 equations, 3 figures)

This paper contains 11 sections, 24 theorems, 71 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Further remarks and open problems
  4. Notation
  5. Critical colour densities for rainbow triangles
  6. Preliminary remarks
  7. Proof strategy
  8. The $\mathbf{F}$-extremal region: the case $\alpha_1\leq \alpha_2 +(1-\sqrt{\alpha_2})^2$
  9. The $\mathbf{H}$-extremal region: the case $\alpha_1\geq \alpha_2 +(1-\sqrt{\alpha_2})^2$
  10. Putting it together: proof of Theorem \ref{['theorem: forcing densities']} and Corollary \ref{['cor: conjecture true']}
  11. Appendix: verifying inequality (\ref{['eq1']})

Key Result

Theorem 1.3

[Keevash, Saks, Sudakov, Verstraëte] If $\mathbf{G}$ is a Gallai $r$-colouring template on $n$ vertices for $n$ sufficiently large, then and these upper bounds are best possible.

Figures (3)

  • Figure 1: The Gallai colouring templates $\mathbf{F}\left(a,b,c\right)$ and $\mathbf{H}\left(a,b,c\right)$ with red (thin lines), green (thick lines) and blue (doubled lines) representing edges in colours $1$, $2$ and $3$ respectively.
  • Figure 2: The regions $\mathcal{R}'_1$ (upper part, in green) and $\mathcal{R}_2$ (lower part, in blue) in the $(\alpha_1, \alpha_2)$ plane.
  • Figure 3: A plot of the function $k(d)$ from \ref{['eq1']} for $d\in [0,1]$

Theorems & Definitions (63)

  • Definition 1.1
  • Definition 1.2: Coloured and rainbow subgraphs
  • Theorem 1.3
  • Theorem 1.4
  • Definition 1.6: Forcing triple
  • Conjecture 1.9: Frankl, Győri, He, Lv, Salia, Tompkins, Varga and Zhu
  • Definition 1.10
  • Remark 1.11
  • Definition 1.12
  • Theorem 1.13
  • ...and 53 more