Two Colors on Straight Line

Points on a straight line are colored in two colors. Prove that it is always possible to find three points of the same color with one being the midpoint of the other two.

Solution

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Copyright © 1996-2017 Alexander Bogomolny

Points on a straight line are colored in two colors. Prove that it is always possible to find three points of the same color with one being the midpoint of the other two.

For convenience, name the colors red and blue.

If the whole line is colored with a single color, there is nothing to prove. In any event, there are two points - A and B - of the same color, say, red. Let C be their midpoint. If C is red, we are done. So assume it is blue.

five points - four midpoints - on a line

Mark points D (on the side of A) and E (on the side of B) such that AD = AB = BE. If one of them is red we are done. For example, if D is red then A is the midpoint of DB and the three points all colored red. If both D and E are blue, then we are getting a triple of blue points - D, C, E - with C being the midpoint of DE.

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| |Store|

Copyright © 1996-2017 Alexander Bogomolny

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