(i) Given a positive integer $k$, show that there exists a positive integer $n$ such that, whenever the edges of the complete graph $K_{n}$ are coloured with $k$ colours, there exists a monochromatic triangle.

Denote the least such $n$ by $f(k)$. Show that $f(k) \leqslant 3 \cdot k !$ for all $k$.

(ii) You may now assume that $f(2)=6$ and $f(3)=17$.

Let $H$ denote the graph of order 4 consisting of a triangle together with one extra edge. Given a positive integer $k$, let $g(k)$ denote the least positive integer $n$ such that, whenever the edges of the complete graph $K_{n}$ are coloured with $k$ colours, there exists a monochromatic copy of $H$. By considering the edges from one vertex of a monochromatic triangle in $K_{7}$, or otherwise, show that $g(2) \leqslant 7$. By exhibiting a blue-yellow colouring of the edges of $K_{6}$ with no monochromatic copy of $H$, show that in fact $g(2)=7$.

What is $g(3) ?$ Justify your answer.