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.