Let \(\textbf{G}:=(G_1, G_2, G_3)\) be a triple of graphs on the same vertex set V of size n. A rainbow triangle in \(\textbf{G}\) 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 \(\textbf{G}\) 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 \(\textbf{G}\) 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, Győri, He, Lv, Salia, Tompkins, Varga and Zhu.

令\(\textbf{G}:=(G_1, G_2, G_3)\)为大小为n的同一顶点集V上的图的三元组。\(\textbf{G}\)中的彩虹三角形是边的三元组\((e_1, e_2, e_3)\)，其中每个i为\(e_i\in G_i\)且\(\{e_1, e_2, e_3\}\)在V中形成一个三角形。不包含彩虹三角形的三元组\(\textbf{G}\)也称为 Gallai 着色模板，是极值组合学中广泛研究的一类对象。在目前的工作中，我们完全确定边缘密度集\((\alpha _1, \alpha _2, \alpha _3)\)使得如果\(\vert E(G_i)\vert > \alpha _i n^2 \)对于每个i和n足够大，则\(\textbf{G}\)必须包含一个彩虹三角形。这解决了 Aharoni、DeVos、de la Maza、Montejanos 和 Šámal 提出的问题，概括了先前关于极值 Gallai 着色模板的几个结果，并证明了 Frankl、Győri、He、Lv、Salia、Tompkins、Varga 和 Zhu 的最新猜想。