论文标题
不可避免的超图
Unavoidable hypergraphs
论文作者
论文摘要
Chung andErdő在80年代初引起了以下非常自然的问题，此后已多次重复。在所有$ r $ -graphs $ \ mathcal {h} $中，turán数字$ \ text {ex}（n，\ mathcal {h}）$的最低限度是多少？他们的实际重点是一个同等的，甚至更自然的问题，它询问$ r $ - 绘图的最大尺寸是什么是在$ n $ dertices和$ e $ edges上无法避免的任何$ r $ graph的最大尺寸？ 在原始论文中，他们在$ e $的大部分范围内均不为图形解决这个问题。在后续工作中，Chung和Erdős解决了$ 3 $均匀的案例，并将$ 4 $均匀的案例作为自然的下一步。在本文中，我们在40多年来通过渐近解决$ 4 $均匀的案件在40多年的时间里取得了首要的进展，这使我们有一些迹象表明答案一般应该表现。
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 $\text{ex}(n,\mathcal{H})$ 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.
