The chromatic number of the plane: The bounded case

摘要

The chromatic number of the plane is the smallest number of colors needed in order to paint each point of the plane so that no two points (exactly) unit distance apart are the same color. It is known that seven colors suffice and (at least) four colors are necessary. In order to understand two and three colorings better, it is interesting to see how large a region can be and still be two or three colorable, and how complicated the colorings need to be. This paper gives tight bounds for two and three coloring regions bounded by circles, rectangles, and regular polygons. In particular, a square is 2-colorable if and only the length of a side is ⩽2/5, and is 3-colorable if and only the length of a side is ⩽8/65.