(a) Define ex $(n, H)$ where $H$ is a graph with at least one edge and $n \geqslant|H|$. Show that, for any such $H$, the $\operatorname{limit}_{n \rightarrow \infty} \lim (n, H) /\left(\begin{array}{c}n \\ 2\end{array}\right)$ exists.

[You may not assume the Erdős-Stone theorem.]

(b) State the Erdős-Stone theorem. Use it to deduce that if $H$ is bipartite then $\lim _{n \rightarrow \infty} \operatorname{ex}(n, H) /\left(\begin{array}{l}n \\ 2\end{array}\right)=0 .$

(c) Let $t \geqslant 2$. Show that ex $\left(n, K_{t, t}\right)=O\left(n^{2-\frac{1}{t}}\right)$.

We say $A \subset \mathbb{Z}_{n}$ is nice if whenever $a, b, c, d \in A$ with $a+b=c+d$ then either $a=c, b=d$ or $a=d, b=c$. Let $f(n)=\max \left\{|A|: A \subset \mathbb{Z}_{n}, A\right.$ is nice $\}$. Show that $f(n)=O(\sqrt{n}) .$

$\left[\mathbb{Z}_{n}\right.$ denotes the set of integers modulo $n$, i.e. $\mathbb{Z}_{n}=\{0,1,2, \ldots, n-1\}$ with addition modulo $n .]$