In a coloring of a set of points P with respect to a family of geometric 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 0.368 ) colors always suffice, and Ω(log n/ log 2 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.
There is however a proper O(log n)-coloring with respect to rectangles that contain at least 3 points from S REF .
On coloring points with respect to rectangles
J. Comb. Theory, Ser. A
J. Comb. Theory, Ser. A
Mathematics Computer Science