# 11月29日　湖南大学彭岳建教授学术报告

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 hypergraph on $n$ 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 {\em Lagrangian 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 and Tur\'{a}n numbers of their extensions have been studied actively.

The Lagrangian density of an $r$-uniform hypergraph $H$ is the 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 $\pi_{\lambda}(H)\ge r!\lambda{(K_{t-1}^r)}$, 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 result of 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.