# Difference between revisions of "Hadwiger-Nelson problem"

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Line 10: | Line 10: | ||

| 7 | | 7 | ||

| 11 | | 11 | ||

− | | Two 60-120-60-120 rhombi with a common vertex, | + | | Two 60-120-60-120 rhombi with a common vertex, with one pair of sharp vertices coincident and the other joined |

| Not 3-colorable | | Not 3-colorable | ||

|- | |- | ||

Line 21: | Line 21: | ||

| [https://arxiv.org/abs/1804.02385 J] | | [https://arxiv.org/abs/1804.02385 J] | ||

| 31 | | 31 | ||

− | | | + | | 72 |

| Contains 13 copies of H | | Contains 13 copies of H | ||

| Has essentially six 4-colorings in which no H has a monochromatic triangle | | Has essentially six 4-colorings in which no H has a monochromatic triangle | ||

Line 27: | Line 27: | ||

| [https://arxiv.org/abs/1804.02385 K] | | [https://arxiv.org/abs/1804.02385 K] | ||

| 61 | | 61 | ||

− | | | + | | 150 |

| Contains 2 copies of J | | Contains 2 copies of J | ||

− | | In all 4-colorings | + | | In all 4-colorings lacking an H with a monochromatic triangle, all pairs of vertices at distance 4 are monochromatic |

|- | |- | ||

| [https://arxiv.org/abs/1804.02385 L] | | [https://arxiv.org/abs/1804.02385 L] | ||

| 121 | | 121 | ||

− | | | + | | 301 |

| Contains two copies of K and 52 copies of H | | Contains two copies of K and 52 copies of H | ||

− | | | + | | All 4-colorings contain an H with a monochromatic triangle |

|- | |- | ||

| [https://arxiv.org/abs/1804.02385 T] | | [https://arxiv.org/abs/1804.02385 T] | ||

Line 41: | Line 41: | ||

| 15 | | 15 | ||

| Contains one Moser spindle and useful symmetry | | Contains one Moser spindle and useful symmetry | ||

− | | | + | | Three vertices form an equilateral triangle |

|- | |- | ||

| [https://arxiv.org/abs/1804.02385 U] | | [https://arxiv.org/abs/1804.02385 U] | ||

| 15 | | 15 | ||

− | | | + | | 33 |

+ | | Three copies of T at 120-degree rotations | ||

| Contains three Moser spindles | | Contains three Moser spindles | ||

− | |||

|- | |- | ||

| [https://arxiv.org/abs/1804.02385 V] | | [https://arxiv.org/abs/1804.02385 V] | ||

| 31 | | 31 | ||

− | | | + | | 30 |

| Unit vectors at angles consistent with three interlocking Moser spindles | | Unit vectors at angles consistent with three interlocking Moser spindles | ||

| | | | ||

Line 57: | Line 57: | ||

| [https://arxiv.org/abs/1804.02385 W] | | [https://arxiv.org/abs/1804.02385 W] | ||

| 301 | | 301 | ||

− | | | + | | 1230 |

| Cartesian product of V with itself, minus vertices at \sqrt(3) from the centre | | Cartesian product of V with itself, minus vertices at \sqrt(3) from the centre | ||

| | | | ||

Line 63: | Line 63: | ||

| [https://arxiv.org/abs/1804.02385 M] | | [https://arxiv.org/abs/1804.02385 M] | ||

| 1345 | | 1345 | ||

− | | | + | | 8268 |

− | | Cartesian product of W and H | + | | Cartesian product of W and H |

+ | | All 4-colorings have a monochromatic triangle in the central copy of H | ||

+ | | | ||

|- | |- | ||

| [https://arxiv.org/abs/1804.02385 N] | | [https://arxiv.org/abs/1804.02385 N] | ||

| 20425 | | 20425 | ||

+ | | 151311 | ||

+ | | Contains 52 copies of M arranged around the H-copies of L | ||

+ | | Not 4-colorable | ||

| | | | ||

− | | | + | |- |

+ | | [https://arxiv.org/abs/1804.02385 G] | ||

+ | | 1581 | ||

+ | | 7877 | ||

+ | | N "shrunk" by stepwise deletions and replacements of vertices | ||

+ | | Not 4-colorable | ||

|} | |} | ||

## Revision as of 10:48, 14 April 2018

The Chromatic Number of the Plane (CNP) is the chromatic number of the graph whose vertices are elements of the plane, and two points are connected by an edge if they are a unit distance apart. The Hadwiger-Nelson problem asks to compute CNP. The bounds [math]4 \leq CNP \leq 7[/math] are classical; recently [deG2018] it was shown that [math]CNP \geq 5[/math]. This is achieved by explicitly locating finite unit distance graphs with chromatic number at least 5.

## Contents

## Notable unit distance graphs

Name | Number of vertices | Number of edges | Structure | Colorings | |
---|---|---|---|---|---|

Moser spindle | 7 | 11 | Two 60-120-60-120 rhombi with a common vertex, with one pair of sharp vertices coincident and the other joined | Not 3-colorable | |

H | 7 | 12 | Vertices and center of a hexagon | Has essentially four 4-colorings, two of which contain a monochromatic triangle | |

J | 31 | 72 | Contains 13 copies of H | Has essentially six 4-colorings in which no H has a monochromatic triangle | |

K | 61 | 150 | Contains 2 copies of J | In all 4-colorings lacking an H with a monochromatic triangle, all pairs of vertices at distance 4 are monochromatic | |

L | 121 | 301 | Contains two copies of K and 52 copies of H | All 4-colorings contain an H with a monochromatic triangle | |

T | 9 | 15 | Contains one Moser spindle and useful symmetry | Three vertices form an equilateral triangle | |

U | 15 | 33 | Three copies of T at 120-degree rotations | Contains three Moser spindles | |

V | 31 | 30 | Unit vectors at angles consistent with three interlocking Moser spindles | ||

W | 301 | 1230 | Cartesian product of V with itself, minus vertices at \sqrt(3) from the centre | ||

M | 1345 | 8268 | Cartesian product of W and H | All 4-colorings have a monochromatic triangle in the central copy of H | |

N | 20425 | 151311 | Contains 52 copies of M arranged around the H-copies of L | Not 4-colorable | |

G | 1581 | 7877 | N "shrunk" by stepwise deletions and replacements of vertices | Not 4-colorable |

## Blog posts and other online forums

- Has there been a computer search for a 5-chromatic unit distance graph?, Juno, Apr 16, 2016.
- The chromatic number of the plane is at least 5, Jordan Ellenberg, Apr 9 2018.
- Aubrey de Grey: The chromatic number of the plane is at least 5, Gil Kalai, Apr 10 2018.
- Polymath proposal: finding simpler unit distance graphs of chromatic number 5, Aubrey de Grey, Apr 10 2018.
- The chromatic number of the plane is at least 5, Dustin Mixon, Apr 10, 2018.
- Amazing progress on long-standing problems, Scott Aaronson, Apr 11 2018.
- https://dustingmixon.wordpress.com/2018/04/13/the-chromatic-number-of-the-plane-is-at-least-5-part-ii/, Dustin Mixon, Apr 13 2018.

## Code and data

- The 1585-vertex graph in DIMACS format
- A naive translation of 4-colorability of this graph into a SAT problem in DIMACS format
- The vertices of this graph in explicit Sage notation

## Wikipedia

## Bibliography

- [deBE1951] N. G. de Bruijn, P. Erdős, A colour problem for infinite graphs and a problem in the theory of relations, Nederl. Akad. Wetensch. Proc. Ser. A, 54 (1951): 371–373, MR 0046630.

- [deG2018] A. de Grey, The chromatic number of the plane is at least 5, arXiv:1804.02385
- [H1945] H. Hadwiger, Uberdeckung des euklidischen Raum durch kongruente Mengen, Portugaliae Math. 4 (1945), 238–242.
- [MM1961] L. Moser and M. Moser, Solution to Problem 10, Can. Math. Bull. 4 (1961), 187–189.
- [P1998] D. Pritikin, All unit-distance graphs of order 6197 are 6-colorable, Journal of Combinatorial Theory, Series B 73.2 (1998): 159-163.
- [S2008] A. Soifer, The Mathematical Coloring Book, Springer, 2008, ISBN-13: 978-0387746401.