Suppose that we have n≥3 distinct colours. Let f(n) be the greatest integer with the property that every side and every diagonal of a convex polygon with f(n) vertices can be coloured with one of n colours in the following way:
(i) At least two colours are used,
(ii) any three vertices of the polygon determine either three segments of the same colour or of three different colours.
Show that f(n) \le (n\minus{}1)^2 with equality for infintely many values of n. geometrygeometric transformationcombinatorics proposedcombinatorics