Let $G _ { 0 } = \left( S _ { 0 } , A _ { 0 } \right)$ be a particular fixed graph with $s _ { 0 } = s _ { G _ { 0 } }$, $a _ { 0 } = a _ { G _ { 0 } }$, $s_0 \geq 2$, $a_0 \geq 1$. Let $\mathcal{C}_0$ be the set of copies of $G_0$ whose vertices are included in $\llbracket 1, n \rrbracket$: $$\mathcal { C } _ { 0 } = \left\{ H \mid H \text { is a copy of } G _ { 0 } \text { and } H = \left( S _ { H } , A _ { H } \right) \text { with } S _ { H } \subset \llbracket 1 , n \rrbracket \right\}$$ Let $S _ { 0 } ^ { \prime }$ be a fixed set of cardinality $s _ { 0 }$. We denote by $c _ { 0 }$ the number of graphs whose vertex set is $S _ { 0 } ^ { \prime }$ and which are copies of $G _ { 0 }$. Express the cardinality of $\mathcal { C } _ { 0 }$ using $c _ { 0 }$ and using a simple upper bound for $c _ { 0 }$, justify that the cardinality of $\mathcal { C } _ { 0 }$ is less than $n ^ { s _ { 0 } }$.
Let $G _ { 0 } = \left( S _ { 0 } , A _ { 0 } \right)$ be a particular fixed graph with $s _ { 0 } = s _ { G _ { 0 } }$, $a _ { 0 } = a _ { G _ { 0 } }$, $s_0 \geq 2$, $a_0 \geq 1$. Let $\mathcal{C}_0$ be the set of copies of $G_0$ whose vertices are included in $\llbracket 1, n \rrbracket$:
$$\mathcal { C } _ { 0 } = \left\{ H \mid H \text { is a copy of } G _ { 0 } \text { and } H = \left( S _ { H } , A _ { H } \right) \text { with } S _ { H } \subset \llbracket 1 , n \rrbracket \right\}$$
Let $S _ { 0 } ^ { \prime }$ be a fixed set of cardinality $s _ { 0 }$. We denote by $c _ { 0 }$ the number of graphs whose vertex set is $S _ { 0 } ^ { \prime }$ and which are copies of $G _ { 0 }$.\\
Express the cardinality of $\mathcal { C } _ { 0 }$ using $c _ { 0 }$ and using a simple upper bound for $c _ { 0 }$, justify that the cardinality of $\mathcal { C } _ { 0 }$ is less than $n ^ { s _ { 0 } }$.