Two Colors - Three Points

Points in the plane are colored in two colors. Prove that it is always possible to find a monochromatic equilateral triangle, i.e., three points of the same color with all pairwise distances equal.

Solution

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Copyright © 1996-2018 Alexander Bogomolny

Points in the plane are colored in two colors. Prove that it is always possible to find a monochromatic equilateral triangle, i.e., three points of the same color with all pairwise distances equal.

For convenience, name the colors red and blue.

Start with finding three collinear points A, B, C of the same color. Add points D, E, F as shown:

midlines in an equlateral triangle

There are 1 big and 4 small equilateral triangles. Assume A, B, C are red. If one of D or E is red, we are done. Otherwise, they both are blue. Then if F is red, triangle ACF is monochromatic. Otherwise, if F is blue, triangle DEF is monochromatic.

The same argument works for triangles of any shape, since the midlines of a triangle subdivide it into four smaller triangles of the same shape.

References

  1. R. B. J. T. Allenby, A. Slomson, How to Count: An Introduction to Combinatorics, CRC Press, 2011 (2nd edition)
  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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