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 $(\alpha_1, \alpha_2, \alpha_3)$ such that if $\vert E(G_i)\vert> \alpha_i n^2$ 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 \v{S}\'amal, generalises several previous results on extremal Gallai colouring templates, and proves a recent conjecture of Frankl, Gy\"ori, He, Lv, Salia, Tompkins, Varga and Zhu.

翻译：设 $\mathbf{G}:=(G_1, G_2, G_3)$ 为同一顶点集 $V$（$|V|=n$）上的图三元组。若存在三条边 $(e_1, e_2, e_3)$ 满足 $e_i\in G_i$ 且 $\{e_1, e_2, e_3\}$ 构成 $V$ 上的三角形，则称该三元组为 $\mathbf{G}$ 中的彩虹三角形。不含彩虹三角形的三元组 $\mathbf{G}$（即Gallai着色模板）是极值组合学中广泛研究的对象。本文完整刻画了边密度 $(\alpha_1, \alpha_2, \alpha_3)$ 的集合：当每个 $G_i$ 满足 $\vert E(G_i)\vert> \alpha_i n^2$ 且 $n$ 足够大时，$\mathbf{G}$ 必然包含彩虹三角形。这一结果解决了Aharoni、DeVos、de la Maza、Montejanos与Šámal提出的问题，推广了关于极值Gallai着色模板的若干已有结论，并证明了Frankl、Győri、He、Lv、Salia、Tompkins、Varga与Zhu近期提出的猜想。