论文标题
三角形在没有两分悬浮液的图形中
Triangles in graphs without bipartite suspensions
论文作者
论文摘要
给定图表$ t $和$ h $，广义的Turán数字EX $（N，T，H）$是$ n $ vertex图中$ t $的最大副本数，没有$ h $的副本。 Alon和Shikhelman使用ERD \ H OS的结果确定了$ H $的$ H $大于3的渐近学（N，K_3，H）$，并在$ H $是两位数时证明了几个结果。当$ h $具有色度3时，我们会考虑这个问题。即使对于以下相对简单的3个色谱图，这种特殊情况似乎也很具有挑战性。 图$ h $的悬架$ \ widehat h $是从$ h $获得的图表，它添加了与$ h $的所有顶点相邻的新顶点。当$ h $是一条路径，偶数周期或完整的两部分图时，我们在EX $（N，K_3，\ wideHat {h}）上提供了新的上限和下限。我们使用的主要工具之一是三角去除引理，但目前尚不清楚在不使用拆卸引理的情况下是否可以证明更强的陈述。
Given graphs $T$ and $H$, the generalized Turán number ex$(n,T,H)$ is the maximum number of copies of $T$ in an $n$-vertex graph with no copies of $H$. Alon and Shikhelman, using a result of Erd\H os, determined the asymptotics of ex$(n,K_3,H)$ when the chromatic number of $H$ is greater than 3 and proved several results when $H$ is bipartite. We consider this problem when $H$ has chromatic number 3. Even this special case for the following relatively simple 3-chromatic graphs appears to be challenging. The suspension $\widehat H$ of a graph $H$ is the graph obtained from $H$ by adding a new vertex adjacent to all vertices of $H$. We give new upper and lower bounds on ex$(n,K_3,\widehat{H})$ when $H$ is a path, even cycle, or complete bipartite graph. One of the main tools we use is the triangle removal lemma, but it is unclear if much stronger statements can be proved without using the removal lemma.
