For a fixed positive integer n and an r-uniform hypergraph H, the Turán number ex(n, H) is the maximum number of edges in an H-free r-uniform hypergraph on n vertices, and the Lagrangian density of H is defined as \(\pi _{\lambda }(H)=\sup \{r! \lambda (G) : G \;\text {is an}\; H\text {-free} \;r\text {-uniform hypergraph}\}\), where \(\lambda (G)=\max \{\sum _{e \in G}\prod \limits _{i\in e}x_{i}: x_i\ge 0 \;\text {and}\; \sum _{i\in V(G)} x_i=1\}\) is the Lagrangian of G. For an r-uniform hypergraph H on t vertices, it is clear that \(\pi _{\lambda }(H)\ge r!\lambda {(K_{t-1}^r)}\). Let us say that an r-uniform hypergraph H on t vertices is \(\lambda \)-perfect if \(\pi _{\lambda }(H)= r!\lambda {(K_{t-1}^r)}\). A result of Motzkin and Straus implies that all graphs are \(\lambda \)-perfect. It is interesting to explore what kind of hypergraphs are \(\lambda \)-perfect. A conjecture in [22] proposes that every sufficiently large r-uniform linear hypergraph is \(\lambda \)-perfect. In this paper, we investigate whether the conjecture holds for linear 3-uniform paths. Let \(P_t=\{e_1, e_2, \dots , e_t\}\) be the linear 3-uniform path of length t, that is, \(|e_i|=3\), \(|e_i \cap e_{i+1}|=1\) and \(e_i \cap e_j=\emptyset \) if \(|i-j|\ge 2\). We show that \(P_3\) and \(P_4\) are \(\lambda \)-perfect. Applying the results on Lagrangian densities, we determine the Turán numbers of their extensions.

对于固定的正整数Ñ和- [R -uniform超图ħ，图兰数前（Ñ，  ħ）处于边缘的最大数量ħ -free ř -uniform超图上Ñ顶点和的拉格朗日密度ħ被定义as \（\ pi _ {\ lambda}（H）= \ sup \ {r！\ lambda（G）：G \; \ text {是} \; H \ text {-免费} \; r \ text { -uniform hypergraph} \} \），其中\（\ lambda（G）= \ max \ {\ sum _ {e \ in G} \ prod \ limits _ {i \ in e} x_ {i}：x_i \ ge 0 \; \ text {and} \; \ sum _ {i \ in V（G）} x_i = 1 \} \）是G的拉格朗日数。对于[R-在t个顶点上的均匀超图H，很显然\（\ pi _ {\ lambda}（H）\ ge r！\ lambda {（K_ {t-1} ^ r）} \）。让我们说，一个[R -uniform超图^ h上牛逼的顶点是\（\拉姆达\） -完美的，如果\（\ PI _ {\拉姆达}（H）= R！\拉姆达{（K_ {T-1} ^ R ）} \）。Motzkin和Straus的结果表明，所有图都是\（\ lambda \） -完美的。探索什么样的超图是\（\ lambda \）完美的是很有趣的。[22]中的一个推测提出，每个足够大的r-一致线性超图都是\（\ lambda \）-完善。在本文中，我们研究了猜想是否适用于线性3一致路径。令\（P_t = \ {e_1，e_2，\ dots，e_t \} \）为长度为t的线性3均匀路径，即\（| e_i | = 3 \），\（| e_i \ cap e_ {i + 1} | = 1 \）和\（e_i \ cap e_j = \ emptyset \）如果\（| ij | \ ge 2 \）。我们证明\（P_3 \）和\（P_4 \）是\（\ lambda \） -完美的。将结果应用于拉格朗日密度，我们确定其扩展的图兰数。