Rainbow variations on a theme by Mantel: extremal problems for Gallai colouring templates
Victor Falgas-Ravry, Klas Markstr\"om, Eero R\"aty

TL;DR
This paper characterizes the edge density thresholds in triple graphs that guarantee the existence of rainbow triangles, resolving a key extremal problem in Gallai colouring templates and confirming a recent conjecture.
Contribution
It fully determines the edge density conditions ensuring rainbow triangles in Gallai templates, generalizing previous results and solving a conjecture in extremal combinatorics.
Findings
Established exact edge density thresholds for rainbow triangles.
Resolved a problem posed by previous researchers.
Confirmed a recent conjecture in the field.
Abstract
Let be a triple of graphs on the same vertex set of size . A rainbow triangle in is a triple of edges with for each and forming a triangle in . The triples 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 such that if for each and is sufficiently large, then 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,…
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Topology and Set Theory · Mathematical Dynamics and Fractals
