紧循环的极值数

B. Sudakov, István Tomon
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引用次数: 15

摘要

一个$r$ -统一超图$\mathcal{H}$中的紧循环是一个$\ell\geq r+1$顶点的序列$x_1,\dots,x_{\ell}$,使得所有的$r$ -元组$\{x_{i},x_{i+1},\dots,x_{i+r-1}\}$(带下标模$\ell$)都是$\mathcal{H}$的边。V. Sos的一个老问题,也是由J. Verstraete独立提出的,要求在$n$顶点上的$r$ -一致超图中不存在紧环的最大边数。虽然这是一个非常基本的问题,但直到最近,对于$r\geq 3$这个问题还没有一个好的上界。这里我们证明了答案最多为$n^{r-1+o(1)}$,这与$o(1)$误差项紧密相关。我们的证明是基于在$\mathcal{H}$的线形图中找到鲁棒展开式和一定的密度增量类型参数。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
The Extremal Number of Tight Cycles
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. Sos, also posed independently by J. Verstraete, 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.
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