The following very natural problem was raised by Chung and Erdős in the early 80's and has since been repeated a number of times. What is the minimum of the Turán number $\mathrm{ex}(n,\mathcal{H})$ among all r-graphs $\mathcal{H}$ with a fixed number of edges? Their actual focus was on an equivalent and perhaps even more natural question which asks what is the largest size of an r-graph that can not be avoided in any r-graph on n vertices and e edges?

In the original paper they resolve this question asymptotically for graphs, for most of the range of e. In a follow-up work Chung and Erdős resolve the 3-uniform case and raise the 4-uniform case as the natural next step. In this paper we make first progress on this problem in over 40 years by asymptotically resolving the 4-uniform case which gives us some indication on how the answer should behave in general.

以下非常自然的问题是由 Chung 和 Erdős 在 80 年代初期提出的，此后已多次重复。图兰数的最小值是多少$\mathrm{前任}(n,\mathcal{H})$在所有r图中$\mathcal{H}$具有固定数量的边？他们的实际重点是一个等价的，甚至可能更自然的问题，该问题询问在n个顶点和e 个边上的任何r - graph 中都无法避免的r- graph的最大尺寸是多少？

在原始论文中，他们为图渐近地解决了这个问题，对于e 的大部分范围。在后续工作中，Chung 和 Erdős 解决了 3-uniform 案例并提出了 4-uniform 案例作为自然的下一步。在这篇论文中，我们通过渐近地解决 4-uniform 情况在 40 多年来在这个问题上取得了第一个进展，这给了我们一些关于答案一般应该如何表现的指示。