Generalized Balanced Partitions of Two Sets of Points in the Plane

We consider the following problem. Let n ≥ 2, b ≥ 1 and q ≥ 2 be integers. Let R and B be two disjoint sets of n red points and bn blue points in the plane, respectively, such that no three points of R∪B lie on the same line. Let n = n1 + n2 + · · · + nq be an integer-partition of n such that 1 ≤ ni for every 1 ≤ i ≤ q. Then we want to partition R∪B into q disjoint subsets P1 ∪P2 ∪ · · · ∪Pq that satisfy the following two conditions: (i) conv (Pi)∩ conv (Pj) = ∅ for all 1 ≤ i < j ≤ q, where conv(Pi) denotes the convex hull of Pi; and (ii) each Pi contains exactly ni red points and bni blue points for every 1 ≤ i ≤ q. We shall prove that the above partition exists in the case where (i) 2 ≤ n ≤ 8 and 1 ≤ ni ≤ n/2 for every 1 ≤ i ≤ q, and (ii) n1 = n2 = · · · = nq−1 = 2 and nq = 1.