Let $G _ { 0 } = \left( S _ { 0 } , A _ { 0 } \right)$ be a particular fixed graph with $s _ { 0 } = s _ { G _ { 0 } }$, $a _ { 0 } = a _ { G _ { 0 } }$, $s_0 \geq 2$, $a_0 \geq 1$, and let
$$\omega _ { 0 } = \min _ { \substack { H \subset G _ { 0 } \\ a _ { H } \geq 1 } } \frac { s _ { H } } { a _ { H } }$$
Show that the sequence $\left( k ^ { - \omega _ { 0 } } \right) _ { k \geq 2 }$ is a threshold function for the property $\mathcal { P } _ { n }$ : ``contain a copy of $G_0$''.