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

Abstract

AbstractLet $$\textbf{G}:=(G_1, G_2, G_3)$$ 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 $$\textbf{G}$$ G is a triple of edges $$(e_1, e_2, e_3)$$ ( e 1 , e 2 , e 3 ) with $$e_i\in G_i$$ e i ∈ G i for each i and $$\{e_1, e_2, e_3\}$$ { e 1 , e 2 , e 3 } forming a triangle in V. The triples $$\textbf{G}$$ 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 $$(\alpha _1, \alpha _2, \alpha _3)$$ ( α 1 , α 2 , α 3 ) such that if $$\vert E(G_i)\vert > \alpha _i n^2$$ | E ( G i ) | > α i n 2 for each i and n is sufficiently large, then $$\textbf{G}$$ 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.