(a) Suppose that the edges of the complete graph $K_{6}$ are coloured blue and yellow. Show that it must contain a monochromatic triangle. Does this remain true if $K_{6}$ is replaced by $K_{5}$ ?

(b) Let $t \geqslant 1$. Suppose that the edges of the complete graph $K_{3 t-1}$ are coloured blue and yellow. Show that it must contain $t$ edges of the same colour with no two sharing a vertex. Is there any $t \geqslant 1$ for which this remains true if $K_{3 t-1}$ is replaced by $K_{3 t-2}$ ?

(c) Now let $t \geqslant 2$. Suppose that the edges of the complete graph $K_{n}$ are coloured blue and yellow in such a way that there are a blue triangle and a yellow triangle with no vertices in common. Show that there are also a blue triangle and a yellow triangle that do have a vertex in common. Hence, or otherwise, show that whenever the edges of the complete graph $K_{5 t}$ are coloured blue and yellow it must contain $t$ monochromatic triangles, all of the same colour, with no two sharing a vertex. Is there any $t \geqslant 2$ for which this remains true if $K_{5 t}$ is replaced by $K_{5 t-1}$ ? [You may assume that whenever the edges of the complete graph $K_{10}$ are coloured blue and yellow it must contain two monochromatic triangles of the same colour with no vertices in common.]