Sure. Tile the plane with regular hexagons, spaced so their centers are one unit apart. Oriented, say, so that 2 sides of each hexagon are parallel to the x-axis. Specifically so that each hexagon contains three consecutive of its *open* sides (say the bottom, SE, and NE), and two of its vertices (say the SW and NE ones): o * \ o o / *--o Group the hexagonal tiles into rosettes of 7. Use 7 colors (like ROYGBIV) to color each rosette of 7 hexagons the same way. Then if I haven't screwed up, no two points of the same color can be 1 unit apart. --Dan On 2013-04-02, at 11:04 AM, Joerg Arndt wrote:
Can anyone *disproof*: "For every coloring of the plane with finitely many colors and every d there are always two points of the same color with distance d" ?