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On the Ramsey-Turán Density of Triangles
Combinatorica ( IF 1.0 ) Pub Date : 2021-11-24 , DOI: 10.1007/s00493-021-4340-0
Tomasz Łuczak 1 , Joanna Polcyn 1 , Christian Reiher 2
Affiliation  

One of the oldest results in modern graph theory, due to Mantel, asserts that every triangle-free graph on n vertices has at most ⌊n2/4⌋ edges. About half a century later Andrásfai studied dense triangle-free graphs and proved that the largest triangle-free graphs on n vertices without independent sets of size αn, where 2/5 ≤ α < 1/2, are blow-ups of the pentagon. More than 50 further years have elapsed since Andrásfai’s work. In this article we make the next step towards understanding the structure of dense triangle-free graphs without large independent sets.

Notably, we determine the maximum size of triangle-free graphs G on n vertices with α(G) ≥ 3n/8 and state a conjecture on the structure of the densest triangle-free graphs G with α(G) > n/3. We remark that the case α(G) α n/3 behaves differently, but due to the work of Brandt this situation is fairly well understood.



中文翻译:

关于三角形的 Ramsey-Turán 密度

现代图论中最古老的结果之一,由于 Mantel,断言n个顶点上的每个无三角形图最多有 ⌊ n 2 /4⌋ 条边。大约半个世纪后,Andrásfai 研究了密集无三角形图,并证明了n个顶点上最大的无三角形图没有大小为αn 的独立集合,其中 2/5 ≤ α < 1/2,是五边形的爆炸。自 Andrásfai 的工作已经过去了 50 多年。在本文中,我们朝着理解没有大独立集的密集无三角形图的结构迈出了下一步。

值得注意的是,我们确定了n个顶点上的无三角形图G的最大尺寸,其中α ( G ) ≥ 3 n /8 并陈述了对最密集的无三角形图G结构的猜想,其中α ( G ) > n /3 . 我们注意到情况α ( G ) α n /3 的行为不同,但由于 Brandt 的工作,这种情况很容易理解。

更新日期:2021-11-25
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