论文标题
关于断开超图的Turán数字
On Turán numbers for disconnected hypergraphs
论文作者
论文摘要
我们介绍了Turán问题的以下更简单的变体：给定整数$ n> k> r \ geq 2 $和$ m \ geq 1 $，什么是最小的整数$ t $，其中存在$ r $ r $ runiform-robilgraph带有$ n $ n $ vertices，$ t $ edges，$ t $ t $ t $ connected unconty yy $ k $ k $ k $ s的设置？我们证明了此数量及其限制的一些一般估计，并由$ \ binom {n} {r} $标准化为$ n \ rightarrow \ infty $。此外，当$ k = 5 $，$ r = 3 $和$ m \ geq 2 $时，我们为特定情况提供了一个完整的解决方案。
We introduce the following simpler variant of the Turán problem: Given integers $n>k>r\geq 2$ and $m\geq 1$, what is the smallest integer $t$ for which there exists an $r$-uniform hypergraph with $n$ vertices, $t$ edges and $m$ connected components such that any $k$-subset of the vertex set contains at least one edge? We prove some general estimates for this quantity and for its limit, normalized by $\binom{n}{r}$, as $n\rightarrow \infty$. Moreover, we give a complete solution of the problem for the particular case when $k=5$, $r=3$ and $m\geq 2$.
