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.
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Copyright © 1996-2018 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:
- The circle thus drawn contains a point of the same color as O. If that's the case we are finished.
- 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
Elsewhere, I discuss a similar problem of finding two points of different colors. Do have alook.
- 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
|Contact| |Front page| |Contents| |Coloring Plane|
Copyright © 1996-2018 Alexander Bogomolny
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