A hypergraph $H$ is said to be \emph{linear} if every pair of vertices lies in at most one hyperedge. Given a family $\mathcal{F}$ of $r$-uniform hypergraphs (also called $r$-graphs), an $r$-graph $H$ is said to be \emph{$\mathcal{F}$-free} if it contains no member of $\mathcal{F}$ as a subhypergraph. The \emph{linear Turán number} $ex_r^{\mathrm{lin}}(n,\mathcal{F})$ denotes the maximum number of edges in an $\mathcal{F}$-free linear $r$-graph on $n$ vertices. The crown is a linear $3$-graph obtained from three pairwise disjoint edges by adding an edge that intersects each of them in a distinct vertex. Recently, Gyárfás, Ruszinkó, and Sárközy~[\emph{Linear Turán numbers of acyclic triple systems}, European J.\ Combin.\ (2022)] initiated the study of bounds on the linear Turán number for acyclic $3$-uniform linear hypergraphs, including that of the crown. We extend the notion of a crown by defining a $k$-crown, denoted by $C_{1,k}^r$, to be a linear $r$-graph consisting of one base edge together with $k$ pairwise disjoint edges, each intersecting the base in a distinct vertex. In this paper, we establish an upper bound on $ex_r^{\mathrm{lin}}(n,C_{1,k}^r)$, which in particular improves the recent bound of Zhang, Broersma, and Wang~[\emph{Generalized Crowns in Linear $r$-Graphs}, Electron.\ J.\ Combin.\ (2025)] for all $r \geq 4$, without forbidding any auxiliary configuration. We also note that the cases $k\in\{1,2\}$ correspond to the short linear paths $P_2^r$ and $P_3^r$, and can be treated separately.

翻译：超图 $H$ 被称为\emph{线性的}，若其每对顶点至多同时出现在一条超边中。给定一族 $r$-一致超图（也称为 $r$-图）$\mathcal{F}$，若一个 $r$-图 $H$ 不包含 $\mathcal{F}$ 中任何成员作为子超图，则称其为\emph{$\mathcal{F}$-自由的}。\emph{线性图兰数} $ex_r^{\mathrm{lin}}(n,\mathcal{F})$ 表示在 $n$ 个顶点上不含 $\mathcal{F}$ 中成员的线性 $r$-图的最大边数。皇冠是一种线性 $3$-图，由三条两两不相交的边添加一条与每条边各交于不同顶点的边而构成。近期，Gyárfás、Ruszinkó 和 Sárközy~[\emph{线性无环三元组系统的图兰数}, European J.\ Combin.\ (2022)] 首次研究了无环 $3$-一致线性超图（包括皇冠）的线性图兰数上界。我们将皇冠概念推广，定义 $k$-皇冠（记为 $C_{1,k}^r$）为：由一条基边与 $k$ 条两两不相交的边构成的线性 $r$-图，其中每条附加边与基边交于不同顶点。本文建立了 $ex_r^{\mathrm{lin}}(n,C_{1,k}^r)$ 的一个上界，该结果在不禁止任何辅助配置的条件下，特别改进了 Zhang、Broersma 和 Wang~[\emph{线性 $r$-图中的广义皇冠}, Electron.\ J.\ Combin.\ (2025)] 针对所有 $r \geq 4$ 的最新上界。我们还注意到 $k\in\{1,2\}$ 的情形分别对应于短线性路 $P_2^r$ 和 $P_3^r$，可单独处理。