The Turán number ex(n, H) of a graph H is the maximal number of edges in an H-free graph on n vertices. In 1983, Chung and Erdős asked which graphs H with e edges minimise ex(n, H). They resolved this question asymptotically for most of the range of e and asked to complete the picture. In this paper, we answer their question by resolving all remaining cases. Our result translates directly to the setting of universality, a well-studied notion of finding graphs which contain every graph belonging to a certain family. In this setting, we extend previous work done by Babai, Chung, Erdős, Graham and Spencer, and by Alon and Asodi.

中文翻译：

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普遍和不可避免的图表

图兰数前任(n,H) 的图H是一个中的最大边数H- 免费图表n顶点。1983 年，Chung 和 Erdős 询问哪些图H和e边缘最小化前任(n,H）。他们在大多数范围内渐近地解决了这个问题e并要求完成图片。在本文中，我们通过解决所有剩余案例来回答他们的问题。我们的结果直接转化为普遍性的设置，这是一个经过充分研究的发现包含属于某个家族的每个图的图的概念。在这种情况下，我们扩展了 Babai、Chung、Erdős、Graham 和 Spencer 以及 Alon 和 Asodi 之前所做的工作。