Extending the Continuum of Six-Colorings
Sebastian Pokutta
Technische Universität Berlin
andZuse Institute Berlin
visiting Tokyo University via JSPS
pokutta@math.tu-berlin.de@spokutta

joint work with: Konrad Mundinger, Christoph Spiegel, and Max Zimmer

Discrete Optimization x Machine Learning
July 10th, 2024 · Tokyo, Japan

MATH
Berlin Mathematics Research Center

What is this talk about?
Introduction

New combinatorial constructions to color the plane E2
via machine learning.

Sebastian Pokutta · Six-Colorings

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What is this talk about?
Introduction

New combinatorial constructions to color the plane E2
via machine learning.
Why? Hard open problem and first progress in 30 years for off-diagonal variant.

Sebastian Pokutta · Six-Colorings

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What is this talk about?
Introduction

New combinatorial constructions to color the plane E2
via machine learning.
Why? Hard open problem and first progress in 30 years for off-diagonal variant.
Today: Brief overview of approach, two new constructions, and outlook.

Sebastian Pokutta · Six-Colorings

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What is this talk about?
Introduction

New combinatorial constructions to color the plane E2
via machine learning.
Why? Hard open problem and first progress in 30 years for off-diagonal variant.
Today: Brief overview of approach, two new constructions, and outlook.

Outline• The Hadwiger-Nelson Problem: formulation and status• Our machine learning approach• Results a.k.a. constructions(Hyperlinked) References are not exhaustive; check references contained therein.

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The Hadwiger-Nelson Problem
Problem (Nelson 1950, also: Gardner, Moser, Erdős, Harary, Tutte, ...)
What is the smallest number of colors sufficient for coloring the plane in such a way that no twopoints of the same color are at a unit distance apart?

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The Hadwiger-Nelson Problem
Problem (Nelson 1950, also: Gardner, Moser, Erdős, Harary, Tutte, ...)
What is the smallest number of colors sufficient for coloring the plane in such a way that no twopoints of the same color are at a unit distance apart?
Infinite graph with vertex set E2 and edges {x, y} for any x, y ∈ E2 with ∥x − y∥ = 1⇒ chromatic number of the plane 𝜒(E2 )Sebastian Pokutta · Six-Colorings

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The Hadwiger-Nelson Problem
Problem (Nelson 1950, also: Gardner, Moser, Erdős, Harary, Tutte, ...)
What is the smallest number of colors sufficient for coloring the plane in such a way that no twopoints of the same color are at a unit distance apart?
Infinite graph with vertex set E2 and edges {x, y} for any x, y ∈ E2 with ∥x − y∥ = 1⇒ chromatic number of the plane 𝜒(E2 )TheoremAssuming Axiom of Choice (AoC):[Bruĳn and Erdos, 1951]

Any graph is k-colorable iff every finite subgraph of it is k-colorable.

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The Hadwiger-Nelson Problem
Problem (Nelson 1950, also: Gardner, Moser, Erdős, Harary, Tutte, ...)
What is the smallest number of colors sufficient for coloring the plane in such a way that no twopoints of the same color are at a unit distance apart?
Infinite graph with vertex set E2 and edges {x, y} for any x, y ∈ E2 with ∥x − y∥ = 1⇒ chromatic number of the plane 𝜒(E2 )TheoremAssuming Axiom of Choice (AoC):[Bruĳn and Erdos, 1951]

Any graph is k-colorable iff every finite subgraph of it is k-colorable.

This problem has a long and complicated history...

over 14 pages in [Soifer, 2024]

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History
The Hadwiger-Nelson Problem

p. 24 in [Soifer, 2024]

Sebastian Pokutta · Six-Colorings

3 / 23Lower bounds on 𝜒(E2 )Sebastian Pokutta · Six-Colorings

4 / 23Lower bounds through unit distance graphs
The Hadwiger-Nelson Problem

Find unit distance graphs of large chromatic number.

Sebastian Pokutta · Six-Colorings

5 / 23Lower bounds through unit distance graphs
The Hadwiger-Nelson Problem

Find unit distance graphs of large chromatic number.
Definition
A graph G = (V, E) is a unit distance graph if there exists an embedding f : V → E2 of its
vertices in the plane s.t. ∥f (u) − f (v)∥ = 1 if and only {u, v} ∈ E.

Sebastian Pokutta · Six-Colorings

5 / 23Lower bounds through unit distance graphs
The Hadwiger-Nelson Problem

Find unit distance graphs of large chromatic number.
Definition
A graph G = (V, E) is a unit distance graph if there exists an embedding f : V → E2 of its
vertices in the plane s.t. ∥f (u) − f (v)∥ = 1 if and only {u, v} ∈ E.
A triangle gives a lower bound of 3.

Sebastian Pokutta · Six-Colorings

5 / 23Lower bounds through unit distance graphs
The Hadwiger-Nelson Problem

Find unit distance graphs of large chromatic number.
Definition
A graph G = (V, E) is a unit distance graph if there exists an embedding f : V → E2 of its
vertices in the plane s.t. ∥f (u) − f (v)∥ = 1 if and only {u, v} ∈ E.
A triangle gives a lower bound of 3.
The Moser spindle gives a lower bound of 4.

Sebastian Pokutta · Six-Colorings[Moser and Moser, 1961]

5 / 23Lower bounds through unit distance graphs
The Hadwiger-Nelson Problem

Find unit distance graphs of large chromatic number.
Definition
A graph G = (V, E) is a unit distance graph if there exists an embedding f : V → E2 of its
vertices in the plane s.t. ∥f (u) − f (v)∥ = 1 if and only {u, v} ∈ E.
A triangle gives a lower bound of 3.
The Moser spindle gives a lower bound of 4.

Sebastian Pokutta · Six-Colorings[Moser and Moser, 1961]

5 / 23Lower bounds through unit distance graphs
The Hadwiger-Nelson Problem

Find unit distance graphs of large chromatic number.
Definition
A graph G = (V, E) is a unit distance graph if there exists an embedding f : V → E2 of its
vertices in the plane s.t. ∥f (u) − f (v)∥ = 1 if and only {u, v} ∈ E.
A triangle gives a lower bound of 3.
The Moser spindle gives a lower bound of 4.
Theorem
There is a unit distance graph on 20 425 vertices with chromatic number 5.

Sebastian Pokutta · Six-Colorings[Moser and Moser, 1961][De Grey, 2018]

5 / 23Lower bounds through unit distance graphs
The Hadwiger-Nelson Problem

Find unit distance graphs of large chromatic number.
Definition
A graph G = (V, E) is a unit distance graph if there exists an embedding f : V → E2 of its
vertices in the plane s.t. ∥f (u) − f (v)∥ = 1 if and only {u, v} ∈ E.
A triangle gives a lower bound of 3.
The Moser spindle gives a lower bound of 4.
Theorem
There is a unit distance graph on 20 425 vertices with chromatic number 5.

Sebastian Pokutta · Six-Colorings[Moser and Moser, 1961][De Grey, 2018]

5 / 23Lower bounds through unit distance graphs
The Hadwiger-Nelson Problem

Find unit distance graphs of large chromatic number.
Definition
A graph G = (V, E) is a unit distance graph if there exists an embedding f : V → E2 of its
vertices in the plane s.t. ∥f (u) − f (v)∥ = 1 if and only {u, v} ∈ E.
A triangle gives a lower bound of 3.
The Moser spindle gives a lower bound of 4.
Theorem
There is a unit distance graph on 20 425 vertices with chromatic number 5.

Sebastian Pokutta · Six-Colorings[Moser and Moser, 1961][De Grey, 2018]

5 / 23Lower bounds through unit distance graphs
The Hadwiger-Nelson Problem

Find unit distance graphs of large chromatic number.
Definition
A graph G = (V, E) is a unit distance graph if there exists an embedding f : V → E2 of its
vertices in the plane s.t. ∥f (u) − f (v)∥ = 1 if and only {u, v} ∈ E.
A triangle gives a lower bound of 3.
The Moser spindle gives a lower bound of 4.
Theorem
There is a unit distance graph on 20 425 vertices with chromatic number 5.

Sebastian Pokutta · Six-Colorings[Moser and Moser, 1961][De Grey, 2018]

5 / 23Lower bounds through unit distance graphs
The Hadwiger-Nelson Problem

Find unit distance graphs of large chromatic number.
Definition
A graph G = (V, E) is a unit distance graph if there exists an embedding f : V → E2 of its
vertices in the plane s.t. ∥f (u) − f (v)∥ = 1 if and only {u, v} ∈ E.
A triangle gives a lower bound of 3.
The Moser spindle gives a lower bound of 4.

[Moser and Moser, 1961]

Theorem
There is a unit distance graph on 20 425 vertices with chromatic number 5.

[De Grey, 2018]

Simpler constructions with...
1. 1581 vertices

for detail see [De Grey, 2018]

2. 627 vertices[Exoo and Ismailescu, 2020]

3. 553 vertices (as part of Polymath16)

Marĳn Heule, for details see [Mixon, 2021]

4. 509 vertices (as part of Polymath16)

Jaan Parts, for details see [Mixon, 2021]

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Upper bounds on 𝜒(E2 )Sebastian Pokutta · Six-Colorings

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Upper bounds through colorings
The Hadwiger-Nelson Problem

Explicit colorings g : E2 → [c] := {1, . . . , c}, usually derived through tesselations using
simple polytopal shapes, give5 ≤ 𝜒(E2 ) ≤ ...

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Upper bounds through colorings
The Hadwiger-Nelson Problem

Explicit colorings g : E2 → [c] := {1, . . . , c}, usually derived through tesselations using
simple polytopal shapes, give

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Upper bounds through colorings
The Hadwiger-Nelson Problem

Explicit colorings g : E2 → [c] := {1, . . . , c}, usually derived through tesselations using
simple polytopal shapes, give5 ≤ 𝜒(E2 ) ≤ 9.

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Upper bounds through colorings
The Hadwiger-Nelson Problem

Explicit colorings g : E2 → [c] := {1, . . . , c}, usually derived through tesselations using
simple polytopal shapes, give

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Upper bounds through colorings
The Hadwiger-Nelson Problem

Explicit colorings g : E2 → [c] := {1, . . . , c}, usually derived through tesselations using
simple polytopal shapes, give5 ≤ 𝜒(E2 ) ≤ 8.

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Upper bounds through colorings
The Hadwiger-Nelson Problem

Explicit colorings g : E2 → [c] := {1, . . . , c}, usually derived through tesselations using
simple polytopal shapes, give

Sebastian Pokutta · Six-Colorings

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Upper bounds through colorings
The Hadwiger-Nelson Problem

Explicit colorings g : E2 → [c] := {1, . . . , c}, usually derived through tesselations using
simple polytopal shapes, give

Sebastian Pokutta · Six-Colorings

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Upper bounds through colorings
The Hadwiger-Nelson Problem

Explicit colorings g : E2 → [c] := {1, . . . , c}, usually derived through tesselations using
simple polytopal shapes, give5 ≤ 𝜒(E2 ) ≤ 7.

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Upper bounds through colorings
The Hadwiger-Nelson Problem

Explicit colorings g : E2 → [c] := {1, . . . , c}, usually derived through tesselations using
simple polytopal shapes, give5 ≤ 𝜒(E2 ) ≤ 7.
Question. Can we use computers to find admissible colorings g : E2 → [c], i.e.,x ∈ E2 | ∃y ∈ B1 (x) : g(x) = g(y) = ∅?

... attempts, e.g., via discretization and SAT solvers...

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Upper bounds through colorings
The Hadwiger-Nelson Problem

Explicit colorings g : E2 → [c] := {1, . . . , c}, usually derived through tesselations using
simple polytopal shapes, give5 ≤ 𝜒(E2 ) ≤ 7.
Question. Can we use computers to find admissible colorings g : E2 → [c], i.e.,x ∈ E2 | ∃y ∈ B1 (x) : g(x) = g(y) = ∅?

... attempts, e.g., via discretization and SAT solvers...

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Upper bounds through colorings
The Hadwiger-Nelson Problem

Explicit colorings g : E2 → [c] := {1, . . . , c}, usually derived through tesselations using
simple polytopal shapes, give5 ≤ 𝜒(E2 ) ≤ 7.
Question. Can we use computers to find admissible colorings g : E2 → [c], i.e.,x ∈ E2 | ∃y ∈ B1 (x) : g(x) = g(y) = ∅?

... attempts, e.g., via discretization and SAT solvers...
Idea. Use a parameterized and easily differentiable family g𝜃 : E2 → Δc and find∫arg min E𝜃

B1 (x)g𝜃 (x) · g𝜃 (y) dy x ∈ E2 .

Key Point. Approach is continuous in nature.
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New upper bounds via
machine learning?

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One second recap: Neural Networks
Neural Networks as Colorings

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One second recap: Neural Networks
Neural Networks as Colorings

Here. Simply a parameterized continuous function to model the coloring.

Blackbox

input

heavily parameterizedeasily differentiable
universal approximator

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One second recap: Neural Networks
Neural Networks as Colorings

Here. Simply a parameterized continuous function to model the coloring.

Blackbox

input

heavily parameterizedeasily differentiable
universal approximator

outputTheorem (Universal Approximation Theorem)
Feedforward neural networks with certain activation functions are dense (w.r.t. compact
convergence) in the space of continuous functions.

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Can we improve the upper bound?
Neural Networks as Colorings

Idea. Use gradient descent to train a feedforward network g𝜃 to minimize∫∫

L(𝜃) =[−b,b]×[−b,b]B1 (x)g𝜃 (x) · g𝜃 (y) dy dx

for some reasonable b ∈ R?

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Can we improve the upper bound?
Neural Networks as Colorings

Idea. Use gradient descent to train a feedforward network g𝜃 to minimize∫∫

L(𝜃) =[−b,b]×[−b,b]B1 (x)g𝜃 (x) · g𝜃 (y) dy dx

for some reasonable b ∈ R?
Problem. Still a continuous problem. How to compute?

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Can we improve the upper bound?
Neural Networks as Colorings

Idea. Use gradient descent to train a feedforward network g𝜃 to minimize∫∫

L(𝜃) =[−b,b]×[−b,b]B1 (x)g𝜃 (x) · g𝜃 (y) dy dx

for some reasonable b ∈ R?
Problem. Still a continuous problem. How to compute?
Stochastic (Batch) Gradient Descent. Sample point x(i) ∈ [−b, b] × [−b, b] andy(i) ∈ B1 (x) for i = 1, . . . , m

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Can we improve the upper bound?
Neural Networks as Colorings

Idea. Use gradient descent to train a feedforward network g𝜃 to minimize∫∫

L(𝜃) =[−b,b]×[−b,b]B1 (x)g𝜃 (x) · g𝜃 (y) dy dx

for some reasonable b ∈ R?
Problem. Still a continuous problem. How to compute?
Stochastic (Batch) Gradient Descent. Sample point x(i) ∈ [−b, b] × [−b, b] andy(i) ∈ B1 (x) for i = 1, . . . , m and use thatmˆ 𝜃 L(𝜃) ∇𝜃 L(𝜃) ≈ ∇1 Õ∇𝜃 g𝜃 (x(i) ) · g𝜃 (y(i) ),m
i=1

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Can we improve the upper bound?
Neural Networks as Colorings

Idea. Use gradient descent to train a feedforward network g𝜃 to minimize∫∫

L(𝜃) =[−b,b]×[−b,b]B1 (x)g𝜃 (x) · g𝜃 (y) dy dx

for some reasonable b ∈ R?
Problem. Still a continuous problem. How to compute?
Stochastic (Batch) Gradient Descent. Sample point x(i) ∈ [−b, b] × [−b, b] andy(i) ∈ B1 (x) for i = 1, . . . , m and use thatmˆ 𝜃 L(𝜃) ∇𝜃 L(𝜃) ≈ ∇1 Õ∇𝜃 g𝜃 (x(i) ) · g𝜃 (y(i) ),m
i=1

where ∇𝜃 g𝜃 (x(i) ) · g𝜃 (y(i) ) is easily computed through backpropagation,

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Can we improve the upper bound?
Neural Networks as Colorings

Idea. Use gradient descent to train a feedforward network g𝜃 to minimize∫∫

L(𝜃) =[−b,b]×[−b,b]B1 (x)g𝜃 (x) · g𝜃 (y) dy dx

for some reasonable b ∈ R?
Problem. Still a continuous problem. How to compute?
Stochastic (Batch) Gradient Descent. Sample point x(i) ∈ [−b, b] × [−b, b] andy(i) ∈ B1 (x) for i = 1, . . . , m and use thatmˆ 𝜃 L(𝜃) ∇𝜃 L(𝜃) ≈ ∇1 Õ∇𝜃 g𝜃 (x(i) ) · g𝜃 (y(i) ),m
i=1

where ∇𝜃 g𝜃 (x(i) ) · g𝜃 (y(i) ) is easily computed through backpropagation, to adjust theparameters 𝜃 with an appropriate step size 𝛼k throughˆ 𝜃 L(𝜃).
𝜃k+1 = 𝜃k − 𝛼k ∇⇒ Very flexible approach “Deep Annealing”
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Can we improve the upper bound?
Neural Networks as Colorings

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Unfortunately this coloring was already known...
Neural Networks as Colorings

Theorem
99.985% of the plane can be colored with 6 colors such that no two points of the same color are aunit distance apart.
[Pritikin, 1998, Parts, 2020]

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Unfortunately this coloring was already known...
Neural Networks as Colorings

Theorem
99.985% of the plane can be colored with 6 colors such that no two points of the same color are aunit distance apart.
[Pritikin, 1998, Parts, 2020]
Corollary
Any unit distance graph with chromatic number 7 must have at least 6 992 vertices.
⇒ While coloring was known already maybe on the right track?

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Off-diagonal variant

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Going off-diagonal
Neural Networks as Colorings

If we cannot solve the original problem, we study variants of it:

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Going off-diagonal
Neural Networks as Colorings

If we cannot solve the original problem, we study variants of it:
A c-coloring of the plane has type (d1 , . . . , dc )if color i does not contain any points at distance di .

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Going off-diagonal
Neural Networks as Colorings

If we cannot solve the original problem, we study variants of it:
A c-coloring of the plane has type (d1 , . . . , dc )if color i does not contain any points at distance di .
Problem (Soifer in Nash and Rassias’ Open Problems in Mathematics)
Determine the continuum of six-colorings X6 = {d | (1, 1, 1, 1, 1, d) can be realized}.
[Soifer, 1994a, Nash and Rassias, 2016]

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Going off-diagonal
Neural Networks as Colorings

If we cannot solve the original problem, we study variants of it:
A c-coloring of the plane has type (d1 , . . . , dc )if color i does not contain any points at distance di .
Problem (Soifer in Nash and Rassias’ Open Problems in Mathematics)
Determine the continuum of six-colorings X6 = {d | (1, 1, 1, 1, 1, d) can be realized}.
[Soifer, 1994a, Nash and Rassias, 2016]

Status. Six-colorings exist for:√1. d = 1/ 5√
2. d = 2 − 1
3. Part of family with 0.414 ≈[Soifer, 1992]√√
2 − 1 ≤ d ≤ 1/ 5 ≈ 0.447

Sebastian Pokutta · Six-Colorings[Hoffman and Soifer, 1993, 1996][Hoffman and Soifer, 1996, Soifer, 1994b, 2009]14 / 23

Going off-diagonal
Neural Networks as Colorings

If we cannot solve the original problem, we study variants of it:
A c-coloring of the plane has type (d1 , . . . , dc )if color i does not contain any points at distance di .
Problem (Soifer in Nash and Rassias’ Open Problems in Mathematics)
Determine the continuum of six-colorings X6 = {d | (1, 1, 1, 1, 1, d) can be realized}.
[Soifer, 1994a, Nash and Rassias, 2016]

Status. Six-colorings exist for:√1. d = 1/ 5√
2. d = 2 − 1
3. Part of family with 0.414 ≈[Soifer, 1992]√√
2 − 1 ≤ d ≤ 1/ 5 ≈ 0.447[Hoffman and Soifer, 1993, 1996][Hoffman and Soifer, 1996, Soifer, 1994b, 2009]

Our Deep Annealing approach provides two new colorings leading to...
Theorem
X6 contains the closed interval [0.354, 0.657].
Sebastian Pokutta · Six-Colorings[Mundinger et al., 2024]

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Going off-diagonal
Neural Networks as Colorings

If we cannot solve the original problem, we study variants of it:
A c-coloring of the plane has type (d1 , . . . , dc )if color i does not contain any points at distance di .
Problem (Soifer in Nash and Rassias’ Open Problems in Mathematics)
Determine the continuum of six-colorings X6 = {d | (1, 1, 1, 1, 1, d) can be realized}.
[Soifer, 1994a, Nash and Rassias, 2016]

Status. Six-colorings exist for:√1. d = 1/ 5√
2. d = 2 − 1
3. Part of family with 0.414 ≈[Soifer, 1992]√√
2 − 1 ≤ d ≤ 1/ 5 ≈ 0.447[Hoffman and Soifer, 1993, 1996][Hoffman and Soifer, 1996, Soifer, 1994b, 2009]

Our Deep Annealing approach provides two new colorings leading to...
Theorem
X6 contains the closed interval [0.354, 0.657].

[Mundinger et al., 2024]

Note. We have exact constructions for both colorings.
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First coloring: 0.354 ≤ d ≤ 0.553
Neural Networks as Colorings

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First coloring: components
Neural Networks as Colorings

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Second coloring: 0.418 ≤ d ≤ 0.657
Neural Networks as Colorings

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Second coloring: components
Neural Networks as Colorings

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Is this optimal?
Neural Networks as Colorings

Numerical results showing the percentage of points with some conflict for a given
forbidden distance d in the sixth color found over several runs.
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Under the hood
Neural Networks as Colorings

External Links.
1. From a bad trip to colorings...
2. Example of system output
3. Coloring 1 for different d in last color

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Open Problems and Final Remarks

The underlying optimization approach is very flexible:
1. Can we improve the upper bound of the polychromatic number from 6 to 5?
2. Can we improve the upper bound of the chromatic number of E3 from 15 to 14?
3. Can we apply the same ideas to generate graphons and other limit structures?
4. Can we use adversarial networks when the objective is non-differentiable?

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Open Problems and Final Remarks

The underlying optimization approach is very flexible:
1. Can we improve the upper bound of the polychromatic number from 6 to 5?
2. Can we improve the upper bound of the chromatic number of E3 from 15 to 14?
3. Can we apply the same ideas to generate graphons and other limit structures?
4. Can we use adversarial networks when the objective is non-differentiable?
The key question. Are we still working on the 6-coloring?

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Thank you!

Full paper is available at arxiv.org/abs/2404.05509.

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References I

N. d. Bruĳn and P. Erdos. A colour problem for infinite graphs and a problem in the theory of relations. Indigationes Mathematicae, 13:371–373, 1951.
A. D. De Grey. The chromatic number of the plane is at least 5. arXiv preprint arXiv:1804.02385, 2018.
G. Exoo and D. Ismailescu. The chromatic number of the plane is at least 5: a new proof. Discrete & Computational Geometry, 64(1):216–226, 2020.
I. Hoffman and A. Soifer. Almost chromatic number of the plane. Geombinatorics, 3(2):38–40, 1993.
I. Hoffman and A. Soifer. Another six-coloring of the plane. Discrete Mathematics, 150(1-3):427–429, 1996.
D. Mixon. Polymath16, seventeenth thread: Declaring victory. Polymath16, February 1 2021. Retrieved 16 August 2021.
L. Moser and M. Moser. Solution to problem 10. Canadian Mathematical Bulletin, 4:187–189, 1961.
K. Mundinger, S. Pokutta, C. Spiegel, and M. Zimmer. Extending the Continuum of Six-Colorings. Geombinatorics Quarterly, 5 2024.
J. F. Nash and M. T. Rassias. Open problems in mathematics. Springer, 2016.
J. Parts. What percent of the plane can be properly 5-and 6-colored? arXiv preprint arXiv:2010.12668, 2020.
D. Pritikin. All unit-distance graphs of order 6197 are 6-colorable. Journal of Combinatorial Theory, Series B, 73(2):159–163, 1998.
A. Soifer. A six-coloring of the plane. Journal of Combinatorial Theory, Series A, 61(2):292–294, 1992.
A. Soifer. Six-realizable set x6 . Geombinatorics, III(4):140–145, 1994a.
A. Soifer. An infinite class of six-colorings of the plane. Congressus Numerantium, pages 83–86, 1994b.
A. Soifer. The mathematical coloring book: Mathematics of coloring and the colorful life of its creators. Springer, 2009.
A. Soifer. The New Mathematical Coloring Book. Springer, 2024.

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