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.

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。