A copy of a hypergraph $F$ is called an $F$-copy. Let $K_k^r$ denote the complete $r$-uniform hypergraph whose vertex set is $[k] = \{1, \dots, k\}$ (that is, the edges of $K_k^r$ are the $r$-element subsets of $[k]$). Given an $r$-uniform $n$-vertex hypergraph $H$, the $K_k^r$-isolation number of $H$, denoted by $ι(H, K_k^r)$, is the size of a smallest subset $D$ of the vertex set of $H$ such that the closed neighbourhood $N[D]$ of $D$ intersects the vertex sets of the $K_k^r$-copies contained by $H$ (equivalently, $H-N[D]$ contains no $K_k^r$-copy). In this note, we show that if $2 \leq r \leq k$ and $H$ is connected, then $ι(H, K_k^r) \leq \frac{n}{k+1}$ unless $H$ is a $K_k^r$-copy or $k = r = 2$ and $H$ is a $5$-cycle. This solves a recent problem of Li, Zhang and Ye. The result for $r = 2$ (that is, $H$ is a graph) was proved by Fenech, Kaemawichanurat and the author, and is used to prove the result for any $r$. The extremal structures for $r = 2$ were determined by various authors. We use this to determine the extremal structures for any $r$.

翻译：超图$F$的一个副本称为$F$-副本。令$K_k^r$表示顶点集为$[k] = \{1, \dots, k\}$的完全$r$-一致超图（即$K_k^r$的边是$[k]$的所有$r$元子集）。给定一个$n$顶点的$r$-一致超图$H$，其$K_k^r$-隔离数，记作$ι(H, K_k^r)$，是$H$的顶点集的最小子集$D$的大小，使得$D$的闭邻域$N[D]$与$H$所包含的$K_k^r$-副本的顶点集相交（等价地，$H-N[D]$不包含任何$K_k^r$-副本）。在本注记中，我们证明若$2 \leq r \leq k$且$H$连通，则除非$H$本身是一个$K_k^r$-副本，或者$k = r = 2$且$H$是一个$5$-圈，否则有$ι(H, K_k^r) \leq \frac{n}{k+1}$。这解决了Li、Zhang和Ye最近提出的一个问题。$r = 2$的情形（即$H$为图）的结果由Fenech、Kaemawichanurat和本文作者证明，并用于证明任意$r$的结果。$r = 2$时的极值结构已由多位学者确定。我们利用这一点来确定任意$r$时的极值结构。