Pick any point O. Let it be red. Draw a circle of radius √3d. There are two possibilities. All points on that circle are red or there is a point that is either green or blue. In the first case, the endpoints of any chord of length d will be of the same color (red), thus solving the problem. Otherwise, let there be green point P. Find two points A and B that form two equilateral triangles OAB and PAB of side d. The points are the intersections of the two circle O(d) and P(d), centered at O and P and both with radius d. One of A and B may be either red or green. Together with O or P we would have a monochromatic pair. Otherwise, they both are blue and at distance d and so form a monochromatic pair.