Define the Ramsey number $R^{(r)}(s, t)$. What is the value of $R^{(1)}(s, t)$ ? Prove that $R^{(r)}(s, t) \leqslant 1+R^{(r-1)}\left(R^{(r)}(s-1, t), R^{(r)}(s, t-1)\right)$ holds for $r \geqslant 2$ and deduce that $R^{(r)}(s, t)$ exists.

Show that $R^{(2)}(3,3)=6$ and that $R^{(2)}(3,4)=9$.

Show that $7 \leqslant R^{(3)}(4,4) \leqslant 19$. [Hint: For the lower bound, choose a suitable subset $U$ and colour e red if $|U \cap e|$ is odd.]