On the length of longest alternating paths for multicoloured point sets in convex position

Let image be a set of points in image in general position such that each point is coloured with one of image colours. An alternating path of image is a simple polygonal whose edges are straight line segments joining pairs of elements of image with different colours. In this paper we prove the following: suppose that each colour class has cardinality image and image is the set of vertices of a convex polygon. Then image always has an alternating path with at least image elements. Our bound is asymptotically sharp for odd values of image.