论文标题
大型超图无周期
Large hypergraphs without tight cycles
论文作者
论文摘要
$ r $ - 均匀的长度$ \ ell> r $的紧密周期是一个超图，带有顶点$ v_1，\ dots，v_ \ ell $，并边缘$ \ {v_i，v_i {i+1}，\ dots，v_ {i+r-1} \} $（对于所有$ i $），with y modul $ \。 Sudakov和Tomon表明，对于每个固定的$ r \ geq 3 $，$ r $ - 均匀的超图在$ n $ dertices上，它不包含任何长度的紧密周期，最多只有$ n^{r-1+o（1）} $ hyperedges，但最著名的结构（只有最大的edges）（只有最大的edges）提供了$ω（仅提供$ω（n^n^^r-1}）在本说明中，我们证明，对于每种固定的$ r \ geq 3 $，都有$ r $统一的超图，上面有$ω（n^{r-1} \ log n/\ log n/\ log \ log \ log n）$ edges，这些$ edges不包含紧密的周期，表明$ o（1）$在上限的指数中是$ o（1）$。
An $r$-uniform tight cycle of length $\ell>r$ is a hypergraph with vertices $v_1,\dots,v_\ell$ and edges $\{v_i,v_{i+1},\dots,v_{i+r-1}\}$ (for all $i$), with the indices taken modulo $\ell$. It was shown by Sudakov and Tomon that for each fixed $r\geq 3$, an $r$-uniform hypergraph on $n$ vertices which does not contain a tight cycle of any length has at most $n^{r-1+o(1)}$ hyperedges, but the best known construction (with the largest number of edges) only gives $Ω(n^{r-1})$ edges. In this note we prove that, for each fixed $r\geq 3$, there are $r$-uniform hypergraphs with $Ω(n^{r-1}\log n/\log\log n)$ edges which contain no tight cycles, showing that the $o(1)$ term in the exponent of the upper bound is necessary.
