Author:
Falgas-Ravry Victor,Markström Klas,Räty Eero
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.
Publisher
Springer Science and Business Media LLC