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 $4 \leq CNP \leq 7$ are classical; recently [deG2018] it was shown that $CNP \geq 5$. This is achieved by explicitly locating finite unit distance graphs with chromatic number at least 5.

The Polymath16 project seeks to simplify the graphs used in [deG2018] to establish this lower bound. More precisely, the goals are

• Goal 1: Find progressively smaller 5-chromatic unit-distance graphs.
• Goal 2: Reduce (ideally to zero) the reliance on computer assistance for the proof. Computer assistance was leveraged in [deG2018] to analyze a subgraph of size 397.
• Goal 3: Apply these simpler graphs to inform progress in related areas. For example:
• Find a 6-chromatic unit-distance graph in the plane.
• Improve the corresponding bound in higher dimensions.
• Improve the current record of 383/102 for the fractional chromatic number of the plane.
• Find the smallest unit-distance graph of a given minimum degree (excluding, in some natural way, boring cases like Cartesian products of a graph with a hypercube).

## Notable unit distance graphs

A unit distance graph is a graph that can be realised as a collection of vertices in the plane, with two vertices connected by an edge if they are precisely a unit distance apart. The chromatic number of any such graph is a lower bound for $CNP$; in particular, if one can find a unit distance graph with no 4-colorings, then $CNP \geq 5$. The boldface number of vertices is the current minimal number of vertices of a unit distance graph that is currently known to not be 4-colorable.

$G_1 \oplus G_2$ denotes the Minkowski sum of two unit distance graphs $G_1,G_2$ (vertices in $G_1 \oplus G_2$ are sums of the vertices of $G_1,G_2$). $G_1 \cup G_2$ denotes the union. $\mathrm{rot}(G, \theta)$ denotes $G$ rotated counterclockwise by $\theta$. $\mathrm{trim}(G,r)$ denotes the trimming of $G$ after removing all vertices of distance greater than $r$ from the origin.

Another basic operation is spindling: taking two copies of a graph $G$, gluing them together at one vertex, and rotating the copies so that the two copies of another vertex are a unit distance apart. For instance, the Moser spindle is the spindling of a rhombus graph. If a graph has two vertices forced to be the same color in a k-coloring, then the spindling of that graph at those two vertices is not k-colorable.

Many of the graphs are embeddable in abelian groups or rings, particularly those generated by the complex numbers of unit modulus $\omega_t := \exp( i \arccos( 1 - \tfrac{1}{2t} ))$ for various natural numbers $t$, particularly the Loeschian numbers $1,3,4,7,9,12,\dots$. These numbers arise naturally as the apex angle of a $\sqrt{t}, \sqrt{t}, 1$ isosceles triangle, and the distances $\sqrt{t}$ are the distances that arise in the triangular lattice. The rings $R_n = {\bf Z}[ \omega_{t_1}, \dots, \omega_{t_n}]$, where $t_1,t_2,\dots$ are the Loeschian numbers, seem particularly relevant, thus $R_0 = {\bf Z}, R_1 = {\bf Z}[\omega_1], R_2 = {\bf Z}[\omega_1, \omega_3]$, etc.. Closely related rings are the rings $\overline{R_n}$ generated by the unit vectors in $R_n$ and their inverses.

Note that the square root $\eta = \exp( i \frac{1}{2} \arccos \frac{5}{6} )$ of $\omega_3$ lies in $R_2$, thanks to the identity

$\eta = (\omega_1^4 + \omega_1^5) (\omega_3 - 1).$
Name Number of vertices Number of edges Structure Group 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 ${\bf Z}[\omega_1, \omega_3]$ Not 3-colorable
Golomb graph 10 18 Contains the center and vertices of a hexagon and equilateral triangle ${\bf Z}[\omega_1, \omega_3]$ Not 3-colorable
H 7 12 Vertices and center of a hexagon ${\bf Z}[\omega_1]$ Has essentially four 4-colorings, two of which contain a monochromatic $\sqrt{3}$-triangle. Every 5-coloring has a monochrome $\sqrt{3}$-edge or a monochrome $2$-edge
J 31 72 Contains 13 copies of H ${\bf Z}[\omega_1]$ Has essentially six 4-colorings in which no H has a monochromatic $\sqrt{3}$-triangle
K 61 150 Contains 2 copies of J In all 4-colorings lacking an H with a monochromatic $\sqrt{3}$-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 $\sqrt{3}$-triangle
$L_1$ 97 Has 40 copies of H All 4-colorings contain an H with a monochromatic $\sqrt{3}$-triangle
$L_2$ 120 354 All 4-colorings contain an H with a monochromatic $\sqrt{3}$-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: $T \cup \mathrm{rot}(T, 2\pi/3) \cup \mathrm{rot}(T, 4\pi/3)$
V 31 30 $\{0\} \cup \{ \omega_1^x \omega_3^{y/2}: x=0,\dots,5; y=0,\dots,4\}$ ${\bf Z}[\omega_1,\omega_3]$
$V_1$ 61 60 Union of V and a rotation of V: $V \cup \mathrm{rot}(V, \mathrm{arccos}(7/8))$ ${\bf Z}[\omega_1,\omega_3, \omega_4]$
$V_a$ 25 24 Star graph
$V_b$ 25 24 Star graph
$V_x$ 13 12 Subgraph of $V_a$
$V_z$ Subgraph of $V_a$; shares a line of symmetry with $V_a$
$V_y$ 13 12 Subgraph of $V_b$
$V_A$ 37 36 Unit vectors with angles $i \frac{\pi}{3} + j \mathrm{arccos} \frac{5}{6} + k \mathrm{arccos} \frac{7}{8}$
W 301 1230 Cartesian product of V with itself, minus vertices at more than $\sqrt{3}$ from the centre (i.e. $\mathrm{trim}(V \oplus V, \sqrt{3})$)
$W_1$ Trimmed product of V with itself ($\mathrm{trim}(V \oplus V, 1.95)$)
M 1345 8268 Cartesian product of W and H ($W \oplus H$) ${\bf Z}[\omega_1, \omega_3]$ All 4-colorings have a monochromatic triangle in the central copy of H
$M_1$ 278 Deleting vertices from M while maintaining its restriction on H All 4-colorings have a monochromatic triangle in the central copy of H
$M_2$ 7075 Sum of H with a trimmed product of $V_1$ with itself Not 4-colorable
N 20425 151311 Contains 52 copies of M arranged around the H-copies of L ${\bf Z}[\omega_1, \omega_3, \omega_4, \omega_{16}]$ Not 4-colorable
$G_0$ 1585 7909 N "shrunk" by stepwise deletions and replacements of vertices Not 4-colorable
G 1581 7877 Deleting 4 vertices from $G_0$ Not 4-colorable
$G_1$ 1577 Deleting 8 vertices from $G_0$ Not 4-colorable
$G_2$ 874 4461 Juxtaposing two copies of M and shrinking ${\bf Z}[\omega_1, \omega_3, \omega_4]$ Not 4-colorable
$G_3$ 826 4273 Not 4-colorable
$G_4$ 803 4144 Not 4-colorable
$G_5$ 633 3166 Subgraph of two copies of $V \oplus V \oplus V$ Not 4-colorable
R Union of $W_1$ and a rotated copy of $W_1$
$\mathrm{trim}(R \oplus H, 1.67)$ 2563 Trimmed sum of R and H Not 4-colorable
$V \oplus V \oplus H$ ${\bf Z}[\omega_1, \omega_3, \omega_{64/9}]$ Has two vertices forced to be the same color in a 4-coloring; also no monochromatic $\sqrt{3}$-triangles
$G_{745}$ 745 Subgraph of $V \oplus V \oplus H$ Has two vertices forced to be the same color in a 4-coloring
$V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H$ 3085 Not 4-colorable
$V_a \oplus V_z \oplus H \cup V_b \oplus V_y \oplus H$ 3049 Not 4-colorable
$G_{1951}$ 1951 Trimmed version of $V_a \oplus V_x \oplus H \cup V_b \oplus V_y \oplus H$ Not 4-colorable
$V_A \oplus V_A \oplus V_A$ 6937 44439 ${\bf Z}[\omega_1, \omega_3, \omega_4]$ Not 4-colorable
$G_{40}$ 40 82 Any 4-coloring that avoids a monochromatic $\sqrt{11/3}$-edge has two specific vertices forced to be the same color
$G_{79}$ 79 165 Spindling of $G_{40}$ Any 4-coloring has a monochromatic $\sqrt{11/3}$-edge
$G_{49}$ 49 180 ${\mathbf Q}[\sqrt{3},\sqrt{11}]^2$ Any 4-coloring either has a specific $\sqrt{11/3}$-edge monochromatic, or a monochromatic $1/\sqrt{3}$-triangle
$G_{51}$ 51 Has a specific $1/\sqrt{3}$-triangle which cannot be monochromatic in a 4-coloring of plane
$G_{627}$ 627 2982 Contains $G_{51}$ Has a specific $1/\sqrt{3}$-triangle which cannot be monochromatic in a 4-coloring
$G_{103}$ 103 All 4-colorings contain a monochromatic $2/\sqrt{3}$-edge
regular pentagon with unit side length 5 5 All 4-colorings contain a monochromatic $(\sqrt{5}+1)/2$-edge
regular pentagon with $(\sqrt{5}-1)/2$ side length 5 5 All 4-colorings contain a monochromatic $(\sqrt{5}-1)/2$-edge
$G_{21}$ 21 49 ${\bf Z}[\omega_1,\omega_3]$ Implies $p_2 \geq 1/4$

## Lower bounds

In [P1988], Pritikin proved that every graph with at most 12 vertices is 4-colorable, and every graph with at most 6197 vertices is 6-colorable. Pritikin's bounds are obtained by coloring the plane with k colors and an additional “wild” color such that points of unit distance are both allowed to receive the wild color. If the wild color occupies a small fraction p of the plane, then an exercise in the probabilistic method gives that any fixed unit-distance graph with n vertices enjoys an embedding in the plane that avoids the wild color (and is therefore k-colorable) provided n<1/p. Pritikin’s coloring for the k=4 case cannot be improved without improving on the densest known subset of the plane that avoids unit distances (originally due to Croft in 1967). See this MO thread for additional information. As such, improving the bound in the k=4 case might require a new technique, whereas the k=5 and 6 cases might be amenable to optimization.

A "tile-based" colouring of the plane must have at least 6 colours, as shown by Townsend [Tow2005]; the same proof with a minor error was also derived by Woodall [W1973]. In [T1999], Thomassen showed that any tiling-based 6-coloring would have to be be "unscaleable", i.e. the maximum diameter of a tile and the minimum separation of same-coloured tiles must both be exactly 1 (so that the distance 1 is excluded by suitable colouring of the boundary points).

## Virtual edge

Warning: other definitions have been proposed and the exact definition of this notion is currently under discussion. The clamping of graph G in this section may be interpreted as the devirtualization of all bichromatic virtual edges with length $d$ and chromatic number $4$.

A virtual edge of a unit distance graph $G$ is a distance $d$ with the property that every 4-coloring of $G$ contains a monochromatic pair of vertices of distance exactly $d$. Observe that if a unit distance graph $G$ has a virtual edge at distance $d$, and if there is another unit distance graph $H$ with a pair of vertices at distance $d$ that cannot be monochromatic in a 4-coloring, then we can create a non-4-colorable unit distance graph by "clamping" a copy of $H$ to every virtual edge in $G$.

Known examples of virtual edges include:

## Bichromatic virtual edge

### Definition

If a unit distance graph $H$ exists with a specific pair of vertices which are distance $d$ apart and is bichromatic in all proper $n$-colorings of $H$, then we say that $H$ virtualizes a bichromatic virtual edge with length $d$ and chromatic number $n$.

### Vacuous properties

Considering the graph of the entire plane, all distances correspond to a virtual edge of chromatic number $CNP-1$.

If a bichromatic virtual edge with length $d$ and chromatic number $n$ exists, then a bichromatic virtual edge with length $d$ and chromatic number $n-1$ exists.

### Convenience and devirtualization

Bichromatic virtual edges behave the same as the unit edge(which is a bichromatic virtual edge of every chromatic number), and thus may be used to simplify the construction of graphs.

Recursively replacing bichromatic virtual edges of a graph $G$ with graphs which virtualize the bichromatic virtual edges through clamping produces a unit distance graph $G'$ where $\chi(G')\geq\chi(G)$.

Devirtualizing bichromatic virtual edges of a graph $G$ (i.e. clamping) has no benefit unless nontrivial points of $G$ and $H_0$ overlap or nontrivial points of $H_1$ and $H_0$ overlap, where $H_0$ and $H_1$ are graphs virtualizing bichromatic virtual edges of $G$. Such overlaps may cause $\chi(G')\gt\chi(G)$. If no nontrivial overlaps exist, then $\chi(G')=\max\left(\chi(G),\chi(H_0),\chi(H_1),\dots\right)$.

Because more than 1 graph virtualizes any given bichromatic virtual edge for a fixed length and fixed chromatic number, multiple devirtualizations exist for every finite graph.

### Relation to rings

A devirtualized ring of graph $G$ is a ring which contains all the vertices of a devirtualization of $G$, where the vertices are interpreted as complex numbers.

Use the implied shorthand for arbitrary sets $\mathbb{S}[\left\{a,b,c,\dots\right\}]=\mathbb{S}[a,b,c,\dots]$.

Let the vertices of graph $G$ generate the ring $S_0[T_0]$. Let the bichromatic virtual edges be virtualizable by the graphs $H_1,\dots,H_m$. Let $S_k[T_k]$ be the ring generated by the vertices of $H_k$. Let $S$ be the ring generated by $\bigcup_{k=0}^m S_k$. Let $T=\bigcup_{k=0}^m T_k$. Then $S[T]$ is a devirtualized ring of $G$.

As multiple graphs may virtualize the same virtual edge, careful choice may be required to minimize the set $T$.

### Multiplicative property

Let $H_0$ be a graph virtualizing a bichromatic virtual edge with length $d_0$ and chromatic number $n$. Let $H_1$ be a graph virtualizing a bichromatic virtual edge with length $d_1$ and chromatic number $n$. Create a graph $H_2$ by scaling $H_0$ by a factor of $d_1$. Graph $H_2$ then virtualizes a bichromatic virtual edge length $d_0d_1$ and chromatic number $n$.

### Notable bichromatic virtual edges

Chromatic number Length $d$ Proof
any 1 trivial case
2 $0\leq d\leq 3$ moving the flexible parts of the graph (0,0),(0,1),(0,2),(0,3)
3 $\left\{\left|\frac{(3+\sqrt{-3})k}{2}+\sqrt{-3}j+(-1)^r\right| \mid k,j\in\mathbb{Z}, r\in\mathbb{R}\right\}$ equilateral triangle tiling

### Continuous ranges of bichromatic virtual edges

Flexible graphs might produce continuous ranges of lengths of bichromatic virtual edges of chromatic number $n$ without having a specific pair which is monochrome for every $n$-coloring. Let the range of lengths be on the interval $d_0\leq d\leq d_1$.

If $1 \lt d_1$, then let $k\in\mathbb{Z}^+,d_0^{k+1} \leq d_1^k$. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number $n$ for all lengths larger than $d_0^k$. A finite area allowed to be the same color, the infinite area of the plane, and a finite upper bound on CNP implies $CNP\gt n$.

If $d_0 \lt 1$, then let $k\in\mathbb{Z}^+,d_1^{k+1} \geq d_0^k$. The multiplicative property of bichromatic virtual edges implies the existence of bichromatic virtual edges of chromatic number $n$ for all non-zero lengths smaller than $d_1^k$. Considering a set of $CNP+1$ points all within distance $d_1^k$ of each other implies $CNP\gt n$.

Using a flexible bichromatic virtual edge of chromatic number $n$ and a graph $H$ of chromatic number $n+1$, a flexible a graph $H'$ of chromatic number $n+1$ can be created by scaling $H$ to some arbitrarily large or arbitrarily small size and clamping flexible bichromatic virtual edges of chromatic number $n$ as a replacement of each rigid bichromatic virtual edge of $H$.

$H$ virtualizing a bichromatic virtual edge of chromatic number $n+1$ does not necessarily men the graph $H'$ virtualizes a bichromatic virtual edge.

## Further questions

• What are the independence ratios of the above unit distance graphs?
• What are the fractional chromatic numbers of these graphs?
• What are the Lovasz numbers of these graphs?
• What about the Erdos unit distance graph ($n$ vertices, $n^{1+c/\log\log n}$ edges)?
• Can we use de Grey’s graph to construct unit-distance graphs that are not 5-colorable? To answer this question, we first need to understand how 5-colorings of de Grey’s graph force small collections of vertices to be colored. Varga and Nazgand provide some thoughts along these lines. Even if we can’t stitch together a 6-chromatic unit-distance graph with these ideas, we might be able to apply them to prove that the measurable chromatic number of the plane is at least 6.
• It appears as though the coordinates of our smallest 5-chromatic graph lie in $\mathbb{Q}[\sqrt{3}, \sqrt{5}, \sqrt{11}]$ (see this). If we view the plane as the complex plane, what is the smallest ring that admits a 5-chromatic single-distance graph? Every single-distance graph in the Eistenstein integers and Gaussian integers is 2-chromatic (see this). David Speyer suggests looking at $\mathbb{Z}[\frac{1+\sqrt{-71}}{2}]$ next.

## Code and data

This dropbox folder will contain most of the data and images for the project.

Data:

Code:

Software: