Lagrangian densities of short 3-uniform linear paths and Tur\'an numbers of their extensions

For a fixed positive integer $n$ and an $r$-uniform hypergraph $H$, the Tur'an 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)$ is the Lagrangian of $G$. For an $r$-uniform hypergraph $H$ on $t$ vertices, it is clear that ${\pi }_{\lambda }(H)\ge r!\lambda (K_{t-1}^r)$. We say that an $r$-uniform hypergraph $H$ on $t$ vertices is perfect if $\pi_{\lambda}(H)= r!\lambda{(K_{t-1}^r)}$\mathrm{.}Let$P_t={e_1, e_2, \dots, e_t}$bethelinear$3$-uniformpathoflength$t$,thatis,$|e_i|=3$,$|e_i \cap e_{i+1}|=1$and$e_i \cap e_j=\emptyset$if$|i-j|\ge 2$\mathrm{.}Weshowthat$P_3$and$P_4$ are perfect, this supports a conjecture in \cite{yanpeng} proposing that all $3$-uniform linear hypergraphs are perfect. Applying the results on Lagrangian densities, we determine the Tur'an numbers of their extensions.