Define the binomial random graph $G(n, p)$, where $n \in \mathbb{N}$ and $p \in(0,1)$.

(a) Let $G_{n} \sim G(n, p)$ and let $E_{t}$ be the event that $G_{n}$ contains a copy of the complete graph $K_{t}$. Show that if $p=p(n)$ is such that $p \cdot n^{2 /(t-1)} \rightarrow 0$ then $\mathbb{P}\left(E_{t}\right) \rightarrow 0$ as $n \rightarrow \infty$.

(b) State Chebyshev's inequality. Show that if $p \cdot n \rightarrow \infty$ then $\mathbb{P}\left(E_{3}\right) \rightarrow 1$.

(c) Let $H$ be a triangle with an added leaf vertex, that is

$$H=\left(\left\{x_{1}, \ldots, x_{4}\right\},\left\{x_{1} x_{2}, x_{2} x_{3}, x_{3} x_{1}, x_{1} x_{4}\right\}\right),$$

where $x_{1}, \ldots, x_{4}$ are distinct. Let $F$ be the event that $G_{n} \sim G(n, p)$ contains a copy of $H$. Show that if $p=n^{-0.9}$ then $\mathbb{P}(F) \rightarrow 1$.