论文标题
狭窄周期的极端数量
The extremal number of tight cycles
论文作者
论文摘要
$ r $ rubiform HyperGraph $ \ Mathcal {H} $的紧密周期是$ \ ell \ geq r+1 $ vertices $ x_1，\ dots，x _ {\ ell} $的序列$ \ {x_ {i}，x_ {i+1}，\ dots，x_ {i+r-1} \} $（带有订阅modulo $ \ ell $）是$ \ mathcal {h} $的边缘。 V.Sós的一个旧问题也由J.Verstraëte独立提出，要求在没有紧张周期的$ n $顶点上的$ r $均均匀超图中的最大边缘数量。尽管这是一个非常基本的问题，但直到最近，对于$ r \ geq 3 $，尚无良好的上限。在这里，我们证明答案最多是$ n^{r-1+o（1）} $，它紧密到$ o（1）$错误项。 我们的证明是基于在$ \ Mathcal {H} $的线图中找到强大的扩展程序以及一定的密度增量类型参数。
A tight cycle in an $r$-uniform hypergraph $\mathcal{H}$ is a sequence of $\ell\geq r+1$ vertices $x_1,\dots,x_{\ell}$ such that all $r$-tuples $\{x_{i},x_{i+1},\dots,x_{i+r-1}\}$ (with subscripts modulo $\ell$) are edges of $\mathcal{H}$. An old problem of V. Sós, also posed independently by J. Verstraëte, asks for the maximum number of edges in an $r$-uniform hypergraph on $n$ vertices which has no tight cycle. Although this is a very basic question, until recently, no good upper bounds were known for this problem for $r\geq 3$. Here we prove that the answer is at most $n^{r-1+o(1)}$, which is tight up to the $o(1)$ error term. Our proof is based on finding robust expanders in the line graph of $\mathcal{H}$ together with certain density increment type arguments.
