Rainbow Variations on a Theme by Mantel: Extremal Problems for Gallai Colouring Templates

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.