a subgraph; such a graph is called $H$-*free*. Katona et al. [@KNS] showed that $f(n,H)={{\rm ex}}(n,H)/{n\choose k}$ is a decreasing function of $n$. The limit $\displaystyle \pi(H)=\lim_{n\to \infty} f(n,H)$, which always exists, is called the [*Turán density*]{} of $H$.

For $k=2$, the graph case, Erdős-Stone-Simonovits proved $\pi(G)=1-\frac{1}{\chi(G)-1}$ for any graph $G$ with chromatic number $\chi(G)\geq 3$. If $G$ is bipartite, then ${{\rm ex}}(n,G)=o(n^2)$. The magnitude of ${{\rm ex}}(n,G)$ is unknown for most bipartite graphs $G$.

Let $K^r_k$ denote the complete $r$-graph on $k$ vertices. Turán determined the value of ${{\rm ex}}(n,K^2_{k})$ which implyies that $\pi(K^2_k)=1-\frac{1}{k-1}$ for all $k\geq 3$. However, no Turán density $\pi(K^r_k)$ is known for any $k>r\geq 3$. The most extensively studied case is when $k=4$ and $r=3$. Turán conjectured [@Tu] that $\pi(K_4^3)= 5/9$. Erdős [@Er81] offered \$500 for determining any $\pi(K^r_k)$ with $k>r\geq 3$ and \$1000 for answering it for all $k$ and $r$. The upper bounds for $\pi(K_4^3)$ have been sequentially improved: $0.6213$ (de Caen [@dC94]), $0.5936$ (Chung-Lu [@ChLu]), $0.56167$ (Razborov [@Razborov], using flag algebra method.) There are a few uniform hypergraphs whose Turán density has been determined: the Fano plane [@FS05; @KS05a], expanded triangles [@KS05b], $3$-books, $4$-books [@FMP], $F_5$ [@FrFu83], extended complete graphs [@Pik], etc. 