论文标题
壁炉架上关于主题的彩虹变化：Gallai着色模板的极端问题
Rainbow variations on a theme by Mantel: extremal problems for Gallai colouring templates
论文作者
论文摘要
令$ \ mathbf {g}：=（g_1，g_2，g_3）$为同一顶点$ n $的同一顶点$ v $上的三倍。 $ \ mathbf {g} $中的彩虹三角形是边缘$（e_1，e_2，e_3）$的三倍，每个$ i $和$ i $ in $ e_i \ in g_i $ in $ i $和$ \ {e_1，e_1，e_2，e_2，e_3，e_3 \} $在$ v $中形成三角形。三元$ \ mathbf {g} $不包含彩虹三角形（也称为加莱着色模板），是一类广泛研究的对象，中的对象类别。 在目前的工作中，我们充分确定了边缘密度$（α_1，α_2，α_3）$的集合，如果$ \ vert e（g_i）\ vert>α_in^2 $对于每个$ i $ and $ i $，$ n $足够大，那么$ \ nathbf {g} $必须包含一个彩虹triangle。这解决了Aharoni，Devos，De La Maza，Montejanos和šámal提出的问题，它概括了极端Gallai着色模板的一些先前结果，并证明了Frankl，Györi，He，He，LV，Salia，Salia，Tompkins，Tompkins，Varga，Varga和Zhu的最新猜想。
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 $(α_1, α_2, α_3)$ such that if $\vert E(G_i)\vert> α_i n^2$ 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.
