Colouring Relatives of Intervals on the Plane, II: Intervals and Rays in Two Directions

I: intersection graphs of intervals on the plane; R: intersection graphs of rays on the plane; S: intersection graphs of families of strings on the plane such that the intersection of any two strings is a connected subset of the plane; Im : intersection graphs of intervals on the plane parallel to some m lines; Rm : intersection graphs of rays on the plane parallel to some m lines. In particular, we have proved that for every integer k ≥ 6, g(R, k) = 3 and that g(R2, 5) = 3. Let us remark that the finiteness of g(I, 4) = f (I, 2), g(S, 4) = f (S, 2) is an open problem. In the present paper we are concerned with I2 and R2. Obviously, R2 ⊆ I2 and thus f (R2, k) ≤ f (I2, k) for every k. The main result of this paper states that they are the same.