Three Colors - Bichromatic Lines

Is there a coloring of the plane with three colors such that any straight line is bichromatic, i.e. only contains points of two colors?

Solution

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Copyright © 1996-2018 Alexander Bogomolny

Is there a coloring of the plane with three colors such that any straight line is bichromatic, i.e. only contains points of two colors?

Yes, there is such coloring. Pick an arbitrary point O and color it red. The plane is the union of all the straight lines that pass through O. For any line, choose either green or blue. Color the whole of the lines, except point O, with the selected color.

An arbitrary line in the plane, if it passes through O, is colored with either red/green or red/blue. If a line does not pass through O, it contains only green and blue points.

References

  1. I. Yashchenko, Invitation to a Mathematical Festival, MSRI/AMS, 2013, p 48
  1. Ramsey's Theorem
  2. Party Acquaintances
  3. Ramsey Number R(3, 3, 3)
  4. Ramsey Number R(4, 3)
  5. Ramsey Number R(5, 3)
  6. Ramsey Number R(4, 4)
  7. Geometric Application of Ramsey's Theory
  8. Coloring Points in the Plane and Elsewhere
  9. Two Colors - Two Points
  10. Three Colors - Two Points
  11. Two Colors - All Distances
  12. Two Colors on a Straight Line
  13. Two Colors - Three Points
  14. Three Colors - Bichromatic Lines
  15. Chromatic Number of the Plane
  16. Monochromatic Rectangle in a 2-coloring of the Plane
  17. Two Colors - Three Points on Circle
  18. Coloring a Graph
  19. No Equilateral Triangles, Please

|Contact| |Front page| |Contents| |Coloring Plane|

Copyright © 1996-2018 Alexander Bogomolny

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