set $\bigcup_{i\in R} \binom{[n]}{i}$. For example, $K^{\{k\}}_n$ is the complete $k$-uniform hypergraph. $K^{[k]}_n$ is the non-uniform hypergraph with all possible edges of cardinality at most $k$.

(0,0)–(2,1)–(2,-1)–cycle; at (0,0) \[vertex\_open\] (v1) \[label=above:[1]{}\] ; at (2,1) \[vertex\_open\] (v2) \[label=above:[2]{}\] ; at (2,-1) \[vertex\_open\] (v3) \[label=below:[3]{}\] ; (v1)–(v2); (v1)–(v3); (v2)–(v3); at (1,-2) [Illustration of $K_{3}^{\{2,3\}}$]{} ;

(0,0)–(2,0)–(3.5,2)–cycle; (2,0)–(3.5,2)–(2,4)–cycle; (3.5,2)–(2,4)–(0,4)–cycle; (2,4)–(0,4)–(-1.5,2)–cycle; (0,4)–(-1.5,2)–(0,0)–cycle; (-1.5,2)–(0,0)–(2,0)–cycle;

at (0,0) \[vertex\_open\] (v1) \[label=below:[1]{}\] ; at (2,0) \[vertex\_open\] (v2) \[label=below:[2]{}\] ; at (3.5,2) \[vertex\_open\] (v3) \[label=right:[3]{}\] ; at (2,4) \[vertex\_open\] (v4) \[label=above:[4]{}\] ; at (0,4) \[vertex\_open\] (v5) \[label=above:[5]{}\] ; at (-1.5,2) \[vertex\_open\] (v6) \[label=left:[6]{}\] ; (v1)–(v2)–(v3)–(v4)–(v5)–(v6)–(v1); at (1,-1) [A (tight) cycle $C_{6}^{\{2,3\}}$]{};

Given a family of hypergraphs ${\mathcal{H}}$ with common set of edge-types $R$, we define $$\pi_n({\mathcal{H}}):= \max \left\{h_{n}(G)\colon v(G)=n, G\subseteq K^{R}_{n}, \text{ and } G \mbox{ contains no subgraph in }{\mathcal{H}}\right\}.$$

A hypergraph $G:=G_n^R$ is [*extremal*]{} with respect to the family ${\mathcal{H}}$ if

1.  $G$ contains no subgraph in ${\mathcal{H}}$.

2.  $h_n(G)=\pi_n({\mathcal{H}})$.

The Turán density of ${\mathcal{H}}$ is defined to be $$\begin{aligned}
    \pi(\mathcal{{\mathcal{H}}}):&= \lim_{n\to \infty} \pi_n({\mathcal{H}}) \\
               &= \lim_{n\to \infty} \max \left\{\sum_{F\in E(G)} \frac{1}{\binom{n}{|F|}}\colon v(G)=n, G\subseteq K^{R}_{n}, 
                \text{ and } G \mbox{ contains no subgraph in }{\mathcal{H}}\right\}\end{aligned}$$ when the limit exists; we will soon show that this limit 