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

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 $\mathrm{\mid }E(G_i)\mathrm{\mid }>{\alpha }_i{n}^2$ 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 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.