If each point of the plane is colored red or blue then there are two points of the same color at distance 1 from each other.

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Copyright © 1996-2018 Alexander BogomolnyIf each point of the plane is colored red or blue then there are two points of the same color at distance 1 from each other.

Draw an equilateral triangle with side length 1. The three vertices of the triangle are painted in one of the two colors, implying that some pair is of the same color.

The problem admits a simple extension: all points in the space a painted one of three colors. Prove there are two at distance 1 that are painted with the same color.

There is a by far more powerful assertion that claims the existence of a rectangle whose vertices are of the same color.

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Copyright © 1996-2018 Alexander Bogomolny