(a) Define the Ramsey number $R(s)$. Show that for all integers $s \geqslant 2$ the Ramsey number $R(s)$ exists and that $R(s) \leqslant 4^{s}$.

(b) For any graph $G$, let $R(G)$ denote the least positive integer $n$ such that in any red-blue colouring of the edges of the complete graph $K_{n}$ there must be a monochromatic copy of $G$.

(i) How do we know that $R(G)$ exists for every graph $G$ ?

(ii) Let $s$ be a positive integer. Show that, whenever the edge of $K_{2 s}$ are red-blue coloured, there must be a monochromatic copy of the complete bipartite graph $K_{1, s}$.

(iii) Suppose $s$ is odd. By exhibiting a suitable colouring of $K_{2 s-1}$, show that $R\left(K_{1, s}\right)=2 s$.

(iv) Suppose instead $s$ is even. What is $R\left(K_{1, s}\right) ?$ Justify your answer.