# Chromatic Number of the Plane

If a plane is colored in two colors then, for any unit of length, there is a *monochromatic* pair of points, i.e. a pair of points of the same color, at exactly the unit distance from each other. This is actually true even if the plane is colored with 3 colors.

### Definition

The smallest number of colors needed in a coloring of the plane to ensure that no monochromatic pair is at the unit distance apart is called the **chromatic number** χ of the plane.

From the results just mentioned, χ ≥ 4. This is quite easy to show that

The resulting 18-gon also tiles the plane and leads to the coloring of the plane with 7 colors.

Now it is clear that the tiling can be rescaled to avoid a monochromatic pair for any given unit of length.

### Theorem

4 ≤ χ ≤ 7.

Interestingly, no progress was made on the low estimate until 2018, when the de Grey Graph raised the lower bound to 5. The Parts Graph of 509 vertices is the smallest known 5-chromatic graph.

### Remark

A chromatic number could be (and is) associate with sets of points other than the plane. For example, a *chromatic number of a graph* is the minimum number of colors which are assigned to its vertices so as to avoid *monochromatic edges*, i.e., the edges joining vertices of the same color. It is known that, for a planar graph, the chromatic number is at most 4.

### References

- V. Klee, S. Wagon,
*Old And New Unsolved Problems*, MAA, 1991 - A. Soifer,
*Geometric Etudes in Combinatorial Mathematics*, Springer, 2010 (2nd, expanded 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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Copyright © 1996-2018 Alexander Bogomolny

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