What does it mean to say that a graph $G$ has a $k$-colouring? What are the chromatic number $\chi(G)$ and the independence number $\alpha(G)$ of a graph $G$ ? For each $r \geqslant 3$, give an example of a graph $G$ such that $\chi(G)>r$ but $K_{r} \not \subset G$.

Let $g, k \geqslant 3$. Show that there exists a graph $G$ containing no cycle of length $\leqslant g$ with $\chi(G) \geqslant k$.

Show also that if $n$ is sufficiently large then there is a triangle-free $G$ of order $n$ with $\alpha(G)<n^{0.7}$.