On coloring points with respected to rectangles

In a coloring of a set of points P with respect to a family of regions one requires that in every region containing at least two points from P , not all the points are of the same color. Perhaps the most notorious open case is coloring of n points in the plane with respect to axis-parallel rectangles, for which it is known that O(n) colors always suffice, and Ω(log n/ log log n) colors are sometimes necessary. In this note we give a simple proof showing that every set P of n points in the plane can be colored with O(log n) colors such that every axis-parallel rectangle that contains at least three points from P is non-monochromatic.