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.
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 down. So assume it is blue.
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
- R. B. J. T. Allenby, A. Slomson, How to Count: An Introduction to Combinatorics, CRC Press, 2011 (2nd edition)
- Ramsey's Theorem
- Party Acquaintances
- Ramsey Number R(3, 3, 3)
- Ramsey Number R(4, 3)
- Ramsey Number R(5, 3)
- Ramsey Number R(4, 4)
- Geometric Application of Ramsey's Theory
- Coloring Points in the Plane and Elsewhere
- Two Colors - Two Points
- Three Colors - Two Points
- Two Colors - All Distances
- Two Colors on a Straight Line
- Two Colors - Three Points
- Three Colors - Bichromatic Lines
- Chromatic Number of the Plane
- Monochromatic Rectangle in a 2-coloring of the Plane
- Two Colors - Three Points on Circle
- Coloring a Graph
- No Equilateral Triangles, Please
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