Lagrangian densities of Hypergraphs


Topic: Lagrangian densities of Hypergraphs

Speaker: Professor PengYuejian

Event date: 11/29/2019

Event time: 9:00 am

Venue: Lecture Hall 1506

Sponsor: School of Mathematics and Statistics, Institute of Science and Technology


Given a positive integer $n$ and an$r$-uniform hypergraph $H$, the {\em Tur\'an number}$ex(n, H)$ is the maximum number of edges in an $H$-free $r$-uniform hypergraphon $n$ vertices. The {\emTur\'{a}n density} of $H$ is defined as$\pi(H)=\lim_{n\rightarrow\infty} {ex(n,H) \over {n \choose r } }.$ The {\emLagrangian density } of an $r$-uniform graph $H$ is $\pi_{\lambda}(H)=\sup\{r!\lambda(G):G\;is\;H\text{-}free\}$, where $\lambda(G)$ is the Lagrangian of$G$.  The Lagrangian of a hypergraph has been a useful tool in hypergraph extremal problems.  Recently, Lagrangian densities of hypergraphs andTur\'{a}n numbers of their extensions have been studied actively.

The Lagrangian density of an $r$-uniform hypergraph $H$ isthe same as the Tur\'{a}n density of the extension of $H$. Therefore,these two densities of $H$ equal if every pair of vertices of  $H$ is contained in an edge. For example, to determine the Lagrangian density of$K_4^{3}$ is equivalent to determine the Tur\'an density of $K_4^{3}$. For an$r$-uniform graph $H$ on $t$ vertices, it is clear that $\pi_{\lambda}(H)\ger!\lambda{(K_{t-1}^r)}$, where $K_{t-1}^r$ is the complete $r$-uniform graph on$t-1$ vertices. We say that an $r$-uniform hypergraph $H$ on $t$ vertices is$\lambda$-perfect if $\pi_{\lambda}(H)=r!\lambda{(K_{t-1}^r)}$.  A resultof Motzkin and Straus implies that all graphs are $\lambda$-perfect. It is interesting to explore what kind of hypergraphs are $\lambda$-perfect. We present some open problems and recent results.