Balanced intervals of two stes of points on a line or circle

Abstract

Let n,m, k, h be positive integers such that 1 ≤ n ≤ m, 1 ≤ k ≤ n and 1 ≤ h ≤ m. Then we give a necessary and sufficient condition for every configuration with n red points and m blue points on a line or circle to have an interval containing precisely k red points and h blue points.