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, an $r$-uniform hypergraph $H$ is \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 hyperedges in an $\mathcal{F}$-free linear $r$-uniform hypergraph on $n$ vertices. 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. In this paper, we extend the study of linear Turán number for acyclic systems to higher uniformity. We first give a construction for any linear $r$-uniform tree with $k$ edges that yields the lower bound $ ex_r^{\mathrm{lin}}(n,T_k^r)\ge {n(k-1)}/{r}, $ under mild divisibility and existence assumptions. Next, we study hypertrees with four edges. We prove the exact bound $ ex_r^{\mathrm{lin}}(n,B_4^r)\le {(r+1)n}/{r} $ and characterize the extremal hypergraph class, where $B_4^r$ is formed from $S_3^r$ by appending a hyperedge incident to a degree-one vertex. We also prove the bound $ ex_r^{\mathrm{lin}}(n,E_4^r)\le {(2r-1)n}/{r} $ for the crown $E_4^r$. Finally, we give a construction showing $ ex_r^{\mathrm{lin}}(n,P_4^r)\ge {(r+1)n}/{r} $ under suitable assumptions and conclude with a conjecture on sharp upper bound for $P_4^r$.

翻译：若超图中任意一对顶点至多出现在一个超边中，则称该超图为\emph{线性}超图。给定一族$r$-一致超图$\mathcal{F}$，若一个$r$-一致超图$H$不包含$\mathcal{F}$中任何成员作为子超图，则称$H$是\emph{$\mathcal{F}$-自由}的。\emph{线性Turán数}$ex_r^{\mathrm{lin}}(n,\mathcal{F})$表示在$n$个顶点上、$\mathcal{F}$-自由的线性$r$-一致超图所能包含的最大超边数。Gyárfás、Ruszinkó和Sárközy~[\emph{Linear Turán numbers of acyclic triple systems}, European J.\ Combin.\ (2022)] 开创了对无圈$3$-一致线性超图的线性Turán数界的研究。本文将此研究推广到更高一致性的无圈系统。我们首先为任意具有$k$条边的线性$r$-一致树给出一个构造，在温和的可除性与存在性假设下，得到下界$ ex_r^{\mathrm{lin}}(n,T_k^r)\ge {n(k-1)}/{r} $。其次，我们研究具有四条边的超树。我们证明了精确上界$ ex_r^{\mathrm{lin}}(n,B_4^r)\le {(r+1)n}/{r} $并刻画了极值超图类，其中$B_4^r$是通过在$S_3^r$上添加一个与一度顶点相关联的超边形成的。我们还证明了皇冠$E_4^r$的界$ ex_r^{\mathrm{lin}}(n,E_4^r)\le {(2r-1)n}/{r} $。最后，我们在适当假设下给出一个构造表明$ ex_r^{\mathrm{lin}}(n,P_4^r)\ge {(r+1)n}/{r} $，并以关于$P_4^r$的尖锐上界猜想作结。