For $s \geqslant 2$, let $R(s)$ be the least integer $n$ such that for every 2 -colouring of the edges of $K_{n}$ there is a monochromatic $K_{s}$. Prove that $R(s)$ exists.

For any $k \geqslant 1$ and $s_{1}, \ldots, s_{k} \geqslant 2$, define the Ramsey number $R_{k}\left(s_{1}, \ldots, s_{k}\right)$, and prove that it exists.

Show that, whenever the positive integers are partitioned into finitely many classes, some class contains $x, y, z$ with $x+y=z$.

[Hint: given a finite colouring of the positive integers, induce a colouring of the pairs of positive integers by giving the pair $i j(i<j)$ the colour of $j-i$.]