Let $G _ { 0 } = \left( S _ { 0 } , A _ { 0 } \right)$ be a particular fixed graph with $s _ { 0 } = s _ { G _ { 0 } }$, $a _ { 0 } = a _ { G _ { 0 } }$, $s_0 \geq 2$, $a_0 \geq 1$. We now assume that $\lim _ { n \rightarrow + \infty } \left( n ^ { \omega _ { 0 } } p _ { n } \right) = + \infty$. For $k \in \llbracket 0 , s _ { 0 } \rrbracket$, we denote:
$$\Sigma _ { k } = \sum _ { \substack { \left( H , H ^ { \prime } \right) \in C _ { 0 } ^ { 2 } \\ s _ { H \cap H ^ { \prime } } = k } } \mathbf { P } \left( H \cup H ^ { \prime } \subset G \right)$$
Let $k \in \llbracket 1 , s _ { 0 } \rrbracket$; show that :
$$\Sigma _ { k } \leq \sum _ { H \in \mathcal { C } _ { 0 } } \binom { s _ { 0 } } { k } \binom { n - s _ { 0 } } { s _ { 0 } - k } c _ { 0 } p _ { n } ^ { 2 a _ { 0 } } p _ { n } ^ { - \frac { k } { \omega _ { 0 } } }$$