# 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 © 1996-2018 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

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Copyright © 1996-2018 Alexander Bogomolny