MathNet 1988 — Part 01 Question 7

Suppose that $S$ is a finite set of at least five points in the plane; some are coloured red, the others are coloured blue. No subset of three or more similarly coloured points is collinear. Show that there is a triangle
(i) whose vertices are all the same colour, and
(ii) at least one side of the triangle does not contain a point of the opposite colour.