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The Extremal Number of Tight Cycles
International Mathematics Research Notices ( IF 1 ) Pub Date : 2021-02-08 , DOI: 10.1093/imrn/rnaa396
Benny Sudakov 1 , István Tomon 1
Affiliation  

A tight cycle in an |$r$|-uniform hypergraph |$\mathcal{H}$| is a sequence of |$\ell \geq r+1$| vertices |$x_1,...,x_{\ell }$| such that all |$r$|-tuples |$\{x_{i},x_{i+1},...,x_{i+r-1}\}$| (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)}$|⁠. This is tight up to the |$o(1)$| error term, and it was shown recently by B. Janzer that this error term is indeed needed. Our proof is based on finding robust expanders in the line-graph of |$\mathcal{H}$| together with certain density increment type arguments.

中文翻译:

紧循环的极数

紧周期| $ R $ | 统一超图| $ \ mathcal {H} $ | | $ \ ell \ geq r + 1 $ |的序列 顶点| $ x_1,...,x _ {\ ell} $ | 这样所有| $ r $ | -tuples | $ \ {x_ {i},x_ {i + 1},...,x_ {i + r-1} \} $ | 带下| $ \ ell $ |⁠的模)是| $ \ mathcal {H} $ |⁠的边。同样由J.Verstraëte独立提出的V.Sós的一个旧问题,要求| $ r $ |中的最大边数。| $ n $ |上的一致超图 顶点,没有紧密的循环。尽管这是一个非常基本的问题,但直到最近,还没有人知道该问题的良好上限。| $ r \ geq 3 $ |⁠。在这里,我们证明答案最多为| $ n ^ {r-1 + o(1)} $ |⁠。这与| $ o(1)$ |紧密相关错误项,最近B. Janzer指出确实需要此错误项。我们的证明是基于在| $ \ mathcal {H} $ | 以及某些密度增量类型参数。
更新日期:2021-02-07
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