Two Colors - Two Points

Points in the plane are each colored with one of two colors: red or blue. Prove that, for a given distance d, there always exist two points of the same color at the distance d from each other.

Solution

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

Copyright © 1996-2017 Alexander Bogomolny

Points in the plane are each colored with one of two colors: red or blue. Prove that, for a given distance d, there always exist two points of the same color at the distance d from each other.

Solution 1

Select a point O. With O as the center, draw a circle of radius d. There are just two possibilities:

  1. The circle thus drawn contains a point of the same color as O. If that's the case we are finished.
  2. All points of the circle have a color different from the O's. Then any chord of length d connects two points of the same color.

A question may be asked by a student or suggested by a teacher, Is it always the case that in a circle of radius d there exists a chord of length d? The answer may at first seem obvious but, depending on a student's level, may eventually prove insurmountable. A discussion here may be extremely useful.

For the lower grades, it's possible to simply stipulate existence of such a chord. For more advanced students, a better axiom would be the one that asserts that two circles, each passing through the center of another, intersect at two points.

Solution 2

Consider any equilateral triangle ABC of a given side d. Of the three points A, B, C, two must be of the same color. These two solve the problem.

Remark

Observe that there are colorings of a line with two colors with no monochromatic pait of points at a given distances. For example, the line is the union of half closed, half open segments [n, n+1). Color these segments so that any adjacent pair is colored with different colors. Then no pair of monochromatic points is at distance 1.

Elsewhere, I discuss a similar problem of finding two points of different colors. Do have alook.

  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| |Store|

Copyright © 1996-2017 Alexander Bogomolny

 62024055

Search by google: