The Motzkin-Rabin Theorem says:
If $A$ and $B$ are finite disjoint sets of points in the plane and $A \cup B$ is noncollinear, then there exists a line that contains at least two points from one of the sets and no points from the other set.
If the hypothesis is strengthened to "both $A$ and $B$ are noncollinear", can the conclusion be strengthened from "at least two points" to "exactly two points"?
No. In Böröczky's construction, there are $2m$ points and $m$ ordinary lines (i.e., line containing exactly two points) such that these lines form a perfect matching among the points (i.e., for every point there is exactly one other point with which it forms an ordinary line). Therefore, any coloring where any pair of points on an ordinary line gets a different color is good, provided that no color class is collinear (which is also easy to achieve).
