An Upper Bound on the Linear Tur\'{a}n Number of $k$-Crowns

Abstract

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 Tur\'{a}n 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, Gy\'arf\'as, Ruszink\'o, and S\'ark\"ozy~[\emph{Linear Tur\'an numbers of acyclic triple systems}, European J.\ Combin.\ (2022)] initiated the study of bounds on the linear Tur\'an 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.