Two Colors - All Distances

Points in the plane are each colored with one of two colors: red or blue. The set of distances between the blue points is blue and the set of distances between the red points is red. Prove that either one or the other contains all the positive reals.

Solution

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Copyright © 1996-2018 Alexander Bogomolny

Points in the plane are each colored with one of two colors: red or blue. The set of distances between the blue points is blue and the set of distances between the red points is red. Prove that either one or the other contains all the positive reals.

Solution

Assume to the contrary that the blue set lacks distance b and the red set lacks distance r. Choose a blue point B and form an isosceles triangle ABC with AB = BC = b and AC = r. Then either both A and C are red or one of them is blue. If they are both red we have two red points at distance r. If one of them blue, we have two blue points at distance b.

  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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