Define the Ramsey numbers $R(s, t)$ for integers $s, t \geqslant 2$. Show that $R(s, t)$ exists for all $s, t \geqslant 2$. Show also that $R(s, s) \leqslant 4^{s}$ for all $s \geqslant 2$.

Let $t \geqslant 2$ be fixed. Give a red-blue colouring of the edges of $K_{2 t-2}$ for which there is no red $K_{t}$ and no blue odd cycle. Show, however, that for any red-blue colouring of the edges of $K_{2 t-1}$ there must exist either a red $K_{t}$ or a blue odd cycle.