a copy of $H$, so the number of copies of $H$ in $G$ is at least $\frac{a}{2r}{n\choose m}/{{n-v(H)\choose m-v(H)}}=b {n\choose v(H)}$ where $b:=\frac{a}{2r}{m\choose v(H)}^{-1}$. $\square$

Supersaturation can be used to show that “blowing up” does not change the Turán density $\pi(H)$ just like in the uniform cases.

For any hypergraph $H_n$ and positive integers $s_1,s_2,\ldots, s_n$, the [*blowup*]{} of $H$ is a new hypergraph $(V,E)$, denoted by $H_n(s_1,s_2,\ldots, s_n)$, satisfying

1.  $V:=\sqcup_{i=1}^n V_i$, where $|V_i|=s_i$.

2.  $E=\cup_{F\in {{\rm E}}(H)} \prod_{i\in F} V_i$.

When $s_1=s_2=\cdots=s_n=s$, we simply write it as $H(s)$.

Consider the following simple example. Take $H$ to be the hypergraph with vertex set $[3]$ and edge set $E=\{\{1,2\}, \{1,2,3\}\}$, a chain. Consider the blow-ups $H(2,1,1)$ and $H(1,1,2)$ illustrated below.

(0,1)–(2,0)–(4,0)–cycle; at (0,1) \[vertex\] (v1) \[label=above:[1]{}\] ; at (2,0) \[vertex\] (v2) \[label=below:[2]{}\] ; at (4,0) \[vertex\] (v3) \[label=right:[3]{}\] ; (v1)–(v2); at (2,-2) (empty) [$H$]{};

(0,1)–(2,0)–(4,0)–cycle; (0,-1)–(2,0)–(4,0)–cycle; at (0,1) \[vertex\] (v11) \[label=above:[$v_{1,1}$]{}\] ; at (0,-1) \[vertex\] (v12) \[label=below:[$v_{1,2}$]{}\] ; at (2,0) \[vertex\] (v2) \[label=left:[$v_2$]{}\] ; at (4,0) \[vertex\] (v3) \[label=right:[$v_3$]{}\] ; (v11)–(v2); (v12)–(v2); at (2,-2) [$H(2,1,1)$]{};

(0,0)–(2,0)–(4,1)–cycle; (0,0)–(2,0)–(4,-1)–cycle; at (0,0) \[vertex\] (v1) \[label=above:[$v_{1}$]{}\] ; at (2,0) \[vertex\] (v2) \[label=right:[$v_{2}$]{}\] ; at (4,1) \[vertex\] (v31) \[label=above:[$v_{3,1}$]{}\] ; at (4,-1) \[vertex\] (v32) \[label=below:[$v_{3,2}$]{}\] 