The Turán number of hypergraphs has been studied extensively. Here we deal with a recent direction, the linear Turán number, and restrict ourselves to linear triple systems, a collection of triples on a set of points in which any two triples intersect in at most one point. For a fixed linear triple system $F$, the linear Turán number ${\mathrm{ex}}_L(n,F)$ is the maximum number of triples in a linear triple system with $n$ points that does not contain $F$ as a subsystem.

We initiate the study of the linear Turán number for an acyclic $F$. In this case ${\mathrm{ex}}_L(n,F)$ is linear in $n$ and we aim for good bounds. Since the case of trees is already difficult for graphs (Erdős–Sós conjecture), we concentrate on matchings, paths and small trees.

In case of matchings, where $M_k$ is the set of $k$ pairwise disjoint triples, we prove that for fixed $k$ and large enough $n$, ${\mathrm{ex}}_L(n,M_k)=f(n,k)$ where $f(n,k)$ is the maximum number of triples that can meet $k-1$ points in a linear triple system on $n$ points. This is an analogue of an old result of Erdős on hypergraph matchings. For the $k$-edge linear path $P_k$ we show (extending some standard path increasing methods used for graphs) that ${\mathrm{ex}}_L(n,P_k)\le 1.5kn$ which is probably far from best possible.

Finding ${\mathrm{ex}}_L(n,F)$ relates to difficult problems on Steiner triple systems and interesting even for small trees. For example, for $P_4$, the path with four triples, ${\mathrm{ex}}_L(n,P_4)\le \frac{4n}{3}$ with equality only for disjoint union of affine planes of order 3. On the other hand, for $E_4$, the tree having three pairwise disjoint triples and a fourth one meeting all of them, we have bounds only: $6\lfloor \frac{n-3}{4}\rfloor \le {\mathrm{ex}}_L(n,E_4)\le 2n$.

超图的图兰数已被广泛研究。在这里，我们处理一个最近的方向，线性图兰数，并将自己限制在线性三元组系统，一组点上的三元组集合，其中任意两个三元组最多在一个点上相交。对于固定线性三重系统$F$, 线性图兰数 ${\mathrm{前任}}_{升}(n,F)$ 是线性三元组中的最大三元组数，其中 $n$ 不包含的点 $F$ 作为一个子系统。

我们开始研究非循环的线性图兰数 $F$. 在这种情况下${\mathrm{前任}}_{升}(n,F)$ 是线性的 $n$我们的目标是良好的界限。由于树的情况对于图已经很困难（Erdős-Sós 猜想），我们专注于匹配、路径和小树。

在匹配的情况下，其中 ${米}_{克}$ 是一组 $克$ 成对不相交的三元组，我们证明对于固定 $克$ 并且足够大 $n$, ${\mathrm{前任}}_{升}(n,{米}_{克})=F(n,克)$ 在哪里 $F(n,克)$ 是可以满足的最大三元组数 $克-1$ 线性三重系统中的点 $n$点。这类似于 Erdős 在超图匹配上的旧结果。为了$克$-边线性路径 ${磷}_{克}$ 我们展示（扩展一些用于图形的标准路径增加方法） ${\mathrm{前任}}_{升}(n,{磷}_{克})\le 1.5克n$ 这可能远非最好的。

寻找 ${\mathrm{前任}}_{升}(n,F)$涉及 Steiner 三重系统上的难题，甚至对小树也很有趣。例如，对于${磷}_4$，具有四个三元组的路径， ${\mathrm{前任}}_{升}(n,{磷}_4)\le \frac{4n}{3}$ 仅对于 3 阶仿射平面的不相交并集才相等。另一方面，对于 ${乙}_4$，树具有三个成对不相交的三元组，第四个与所有三元组相遇，我们只有边界： $6\lfloor \frac{n-3}{4}\rfloor \le {\mathrm{前任}}_{升}(n,{乙}_4)\le 2n$.