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 $\lambda (G)$ 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 ${\pi }_{\lambda }(H)=r!\lambda (K_{t-1}^r)$. 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 $e_i\cap e_j=\varnothing$ if $|i-j|\ge 2$. We show that $P_3$ 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 Turán numbers of their extensions.

17 pages, 6 figures. arXiv admin note: text overlap with arXiv:1609.08983; text overlap with arXiv:1510.03461 by other authors

References in corpus (1)

• On a conjecture of Hefetz and Keevash on Lagrangians of intersecting hypergraphs and Turán numbers