Let $s \geqslant 3$. Define the Ramsey number $R(s)$. Show that $R(s)$ exists and that $R(s) \leqslant 4^{s}$.

Show that $R(3)=6$. Show that (up to relabelling the vertices) there is a unique way to colour the edges of the complete graph $K_{5}$ blue and yellow with no monochromatic triangle.

What is the least positive integer $n$ such that the edges of the complete graph $K_{6}$ can be coloured blue and yellow in such a way that there are precisely $n$ monochromatic triangles?