Lagrangian densities of short 3-uniform linear paths and Turán numbers of their extensions

Abstract

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.