What is the minimum number of triangles in a graph of given order and size? Motivated by earlier results of Mantel and Turán, Rademacher solved the first nontrivial case of this problem in 1941. The problem was revived by Erdős in 1955; it is now known as the Erdős–Rademacher problem. After attracting much attention, it was solved asymptotically in a major breakthrough by Razborov in 2008. In this paper, we provide an exact solution for all large graphs whose edge density is bounded away from $1$ , which in this range confirms a conjecture of Lovász and Simonovits from 1975. Furthermore, we give a description of the extremal graphs.

中文翻译：

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具有给定顺序和大小的图形中三角形的确切最小数量

在给定顺序和大小的图中，三角形的最小数量是多少？受 Mantel 和 Turán 早期结果的启发，Rademacher 在 1941 年解决了这个问题的第一个非平凡案例。这个问题在 1955 年由 Erdős 重新提出。它现在被称为 Erdős-Rademacher 问题。在引起广泛关注后，2008 年 Razborov 的重大突破中渐近求解。在本文中，我们为所有边密度有界的大图提供了精确解 $1$ ，在这个范围内证实了 Lovász 和 Simonovits 从 1975 年开始的猜想。此外，我们给出了极值图的描述。