Paper 4, Section II, F
(i) Given a positive integer , show that there exists a positive integer such that, whenever the edges of the complete graph are coloured with colours, there exists a monochromatic triangle.
Denote the least such by . Show that for all .
(ii) You may now assume that and .
Let denote the graph of order 4 consisting of a triangle together with one extra edge. Given a positive integer , let denote the least positive integer such that, whenever the edges of the complete graph are coloured with colours, there exists a monochromatic copy of . By considering the edges from one vertex of a monochromatic triangle in , or otherwise, show that . By exhibiting a blue-yellow colouring of the edges of with no monochromatic copy of , show that in fact .
What is Justify your answer.