# Three Colors - Bichromatic Lines

Is there a coloring of the plane with three colors such that any straight line is bichromatic, i.e. only contains points of two colors?

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Copyright © 1996-2017 Alexander Bogomolny

Is there a coloring of the plane with three colors such that any straight line is bichromatic, i.e. only contains points of two colors?

Yes, there is such coloring. Pick an arbitrary point O and color it red. The plane is the union of all the straight lines that pass through O. For any line, choose either green or blue. Color the whole of the lines, except point O, with the selected color.

An arbitrary line in the plane, if it passes through O, is colored with either red/green or red/blue. If a line does not pass through O, it contains only green and blue points.

### References

- I. Yashchenko,
*Invitation to a Mathematical Festival*, MSRI/AMS, 2013, p 48

- 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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Copyright © 1996-2017 Alexander Bogomolny

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