A Moser-spindle-free 5-chromatic unit distance graph on 2131 vertices in the plane

Jan Kristian Haugland

REVIEW 4 major objections 5 minor 10 references

A Moser-spindle-free 5-chromatic unit distance graph on 2131 vertices in the plane

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper constructs a 2131-vertex unit-distance graph in the plane that needs five colors and contains no Moser spindle, using arcs of a 7-fold symmetric 21-vertex graph.

desk verdict Fresh and explicit arc-based construction, but the headline result rests on a float-verified lemma and uncertified SAT runs; deserves review with artifacts. read the letter →

arxiv 2608.04542 v3 pith:E3RNKHON submitted 2026-08-05 math.CO

Jan Kristian Haugland This is my paper · ORCID

classification math.CO MSC 05C1552C10

keywords unitdistancegraphchromaticnumberoftheplaneHadwiger-NelsonproblemMoserspindle5-chromaticregularheptagonSATsolvercoloring

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper addresses the Hadwiger-Nelson problem—the minimum number of colours needed to colour the plane so that any two points at unit distance get different colours—and aims to show that a 5-chromatic unit distance graph can be built without the Moser spindle, the 7-vertex 4-chromatic graph used in most other constructions. Its starting point is a 21-vertex graph H with sevenfold symmetry; the 84 unit vectors appearing as arcs of H are used as building blocks. From these arcs the author assembles a 740-vertex graph G1 in which two points at distance sqrt(3) must receive different colours in any 4-colouring. Two rotated copies of G1 then force a monochromatic pair, and a scaled spindle copy makes two adjacent vertices monochromatic, so the resulting 2131-vertex graph G3 cannot be 4-coloured. A SAT solver finds a 5-colouring, and a check confirms G3 contains no Moser spindle, giving a Moser-spindle-free 5-chromatic unit distance graph that arises from a described rule.

What carries the argument

The load-bearing object is the 7-fold symmetric unit distance graph H on 21 vertices—a regular heptagon together with two regular heptagrams and seven equilateral triangles—whose 84 directed unit arcs u0,...,u83 are used as translation vectors. Lemma 2.2 is the numerical engine of the paper: it states that any sum of between two and eight of these vectors either has a coordinate larger than a specified threshold epsilon_n or is exactly the zero vector, realized as a combination of closed circuits in H. This makes it possible to identify vertices and paths numerically and to certify which short paths from (0,0) land exactly on (0,sqrt(3)). The final step uses the classical spindle mechanism: take a graph with a known monochromatic pair, place rotated copies so that a second pair is forced to share a colour, then add a scaled copy in which two vertices at unit distance receive that same colour.

What would settle it

Recompute every sum of two to eight of the 84 unit vectors u_s using exact algebraic or interval arithmetic and compare with the thresholds in Lemma 2.2: a single mismatch would change the definition of G1. Independent of that, a direct SAT search for a 4-colouring of the published 2131-vertex graph G3, or an independent subgraph search for the Moser spindle inside G3, would settle the two headline claims.

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Extended reading notes

Core claim

The central discovery is the explicit graph G3 on 2131 vertices and 12530 edges, defined as the union of two rotated copies of G1 together with a scaled copy, which is 5-chromatic and contains no Moser spindle. The graph G1 is the 7-core of a graph assembled from polygonal paths of length at most six from (0,0) to (0,sqrt(3)) using the 84 unit arcs of H; Lemma 2.2 ensures that these paths can be identified reliably by their numerical endpoints. The author shows that (0,0) and (0,sqrt(3)) form a non-monochromatic pair in any 4-colouring of G1, that the two rotated copies force (-1,0) and (1,0) to be monochromatic, and that the final spindle makes two adjacent vertices share a colour, ruling out 4-colourings. A SAT solver quickly finds a 5-colouring, and the author states that G3 does not contain the Moser spindle as a subgraph.

Load-bearing premise

The whole construction leans on Lemma 2.2, whose exhaustive computer search was carried out with standard double-precision floating-point arithmetic; if any small sum of arc vectors was misclassified by rounding, the graph and its colouring properties would not be the ones claimed.

Editorial extensions

If this is right

The existence of this graph shows that the Moser spindle is not a structural prerequisite for non-4-colourability of unit distance graphs in the plane.
Because the construction is described by a small set of vectors and paths rather than by a giant search, it offers a concrete starting point for building larger families of spindle-free high-chromatic graphs.
The numerical Lemma 2.2 is the only non-rigorous link; turning it into an exact certificate would make the 5-chromatic and Moser-spindle-free claims fully verified.
The count of at most six feasible 4-colourings of the generated lattice is a step toward understanding how 4-colourings behave on the full lattice of arc vectors.
This construction provides a benchmark for whether the 1441-vertex record for Moser-spindle-free examples can be pushed further with structured, non-search-based rules.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Varying the shift theta or the thresholds epsilon_n while keeping the same 21-vertex H could produce a family of spindle-free 5-chromatic graphs, possibly with fewer vertices; this is an obvious parametric search the author leaves implicit.
The reliance on double-precision floats in Lemma 2.2 is an implicit invitation to formal verification: an interval-arithmetic or exact-algebraic proof would upgrade the construction from computer-assisted to fully rigorous.
If the six feasible labellings of the lattice are correct, they may transfer to other heptagon-symmetric unit distance graphs and could help bound the chromatic number of the plane from below in a more structural way.
An independent check that G3 really avoids the Moser spindle would close the only part of the headline claim that is stated without proof in the text.

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Signed reviews

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The short version: this is a genuinely different way to build Moser-spindle-free 5-chromatic unit distance graphs, and the construction is explicit enough to be checked in principle. But the check that matters — Lemma 2.2 — is done in double-precision floats with no certificate or code, and the non-4-colorability and Moser-spindle-freeness of the final graph are SAT-solver assertions without proof artifacts. I would not yet cite the headline claim as established, but the paper deserves a serious referee if the author ships verifiable artifacts.

What's actually new: the arc-based framework built from the 7-fold symmetric 21-vertex graph H. The 84 unit vectors from H, the angle theta derived from geometry rather than fitted, and the construction of G1 as a 7-core of a graph induced by short paths between (0,0) and (0,sqrt3) is a clear, structured approach. Proposition 2.1 is proved cleanly with trig identities. Appendix A gives explicit polygonal paths for all 740 vertices of G1, which is a concrete artifact. The paper is also honest: Section 4 is flagged as unfinished, and the author does not oversell the 2131-vertex count as a record.

Soft spots, in order of importance. Lemma 2.2 is load-bearing: it decides when two numerically close vertices of T5 and T6 are actually identical, so the vertex set of G1, hence G3, inherits a float-based exhaustive search. The thresholds (0.045 down to 0.00071) are large enough that the search is probably correct, but 'probably' is not a proof. The paper needs either exact arithmetic, interval/ball arithmetic with rigorous error bounds, or a machine-checked certificate. Second, the claim that G1 is not 4-colorable is verified by CaDiCaL, and the 5-coloring of G3 and Moser-spindle-freeness are similarly asserted. No DRAT certificate or code is provided, so an independent referee cannot confirm without reimplementing the entire pipeline. Third, the identity condition for vertices of T6 uses epsilon_6 for all pairs, which is fine if Lemma 2.2 holds, but again inherits its weakness. Section 4 is a sketch and should not be treated as a result; the author says as much.

Who is this for: people working on Hadwiger-Nelson and unit distance graphs, particularly those interested in construction methods that avoid the Moser spindle. The final graph is larger than the 1441-vertex record, so the value is structural, not record-based.

Recommendation: send to peer review, but condition it on the author providing the construction script, exact or rigorous verification of Lemma 2.2, and SAT certificates or at least a reproducible solver invocation. If those arrive, I'd expect the result to hold.

Referee Report

4 major / 5 minor

Summary. The paper constructs a unit distance graph G3 on 2131 vertices and 12530 edges, claimed to be 5-chromatic and Moser-spindle-free. The construction begins with a 7-fold symmetric unit distance graph H on 21 vertices, derives 84 unit vectors from its arcs, and defines graphs T5 and T6 induced by polygonal paths of length up to 5 and 6 from A=(0,0) to B=(0,√3). A graph G0 is formed from T5 and T6, and its 7-core G1 has 740 vertices; the paper asserts via a SAT solver that A and B are a non-monochromatic pair in every 4-coloring of G1. Rotating and copying G1 gives G2, and a spindle construction yields G3. The non-4-colorability of G3 is deduced from monochromatic pairs, a SAT solver finds a 5-coloring, and a verification asserts that G3 contains no Moser spindle. Section 4 gives an upper bound of 6 on the number of 4-colorings of the lattice generated by the unit vectors.

Significance. If correct, this provides a new, structured construction of a Moser-spindle-free 5-chromatic unit distance graph, complementing the recent record-oriented examples. The geometric identities in Proposition 2.1 are clean, the construction rule is transparent, and Appendix A gives explicit paths covering G1, which aids reproducibility. However, the central claims rest on numerical computations and SAT solver runs that are not certified, so the result is not yet fully established as a mathematical proof.

major comments (4)

[Section 2, Lemma 2.2] The lemma is stated for a set S of indices, but Section 3 applies it to paths i_1...i_n that may repeat vectors. As written, 'S={s_1,...,s_n}' with '0≤s<84 ∀s∈S' suggests distinct elements; if the exhaustive search only checked subsets, the lemma does not cover repeated vectors, which are needed for the path-endpoint argument. The proof also rests on a double-precision exhaustive search, which cannot certify exact equalities of trigonometric sums. This is load-bearing: the identification of vertices of T5 and T6, the counts |V(T5)|=1042 and |V(T6)|=12856, and the definition of G0/G1 all rely on Lemma 2.2. Please provide an exact or interval-arithmetic certificate, and clarify the statement to cover sequences or multisets.
[Section 3, after the definition of G1] The assertions that G1 is not 4-colorable with A and B identified, that G3 admits a 5-coloring, and that G3 is Moser-spindle-free are stated as outputs of a SAT solver or verification without certificates or detailed methodology. These claims are not independently verifiable from the manuscript. Please provide a DRAT certificate for the unsatisfiability, an explicit 5-coloring of G3 (for example, in ancillary material), and a precise description (ideally with a certificate) of the Moser-spindle subgraph check.
[Section 3, paragraph before G3] The statement that among (−1,0), (0,0), (1,0), (−1/2,√3/2), and (1/2,√3/2), the only pair that is not non-monochromatic is (−1,0),(1,0) is asserted without proof. This is load-bearing for the non-4-colorability of G3, since the argument uses that (1,0) is monochromatic with (−1,0). Please provide a derivation from the non-monochromatic pair property of G1 and the rotation formulas.
[Section 4] The upper bound of 6 on the number of 4-colorings of the lattice is conditional on the author's own caveat: 'Note that we have not actually proved that a unit distance vector between two vertices in the lattice must necessarily belong to {u_n} for sufficiently large graph distances.' As written, the claim that there are at most 6 4-colorings is not established, and the finite-chunk SAT verification with virtual edges from G1 does not prove the infinite lattice statement. Please mark this result as conditional or fill the gap.

minor comments (5)

[Section 3, first paragraph] The phrase 'polynomial path' appears twice and should be 'polygonal path'.
[Section 3, definition of V1 and V2] The matrix formulas for V1 and V2 are terse; for readability, explicitly state the center of rotation and verify that the linear map corresponds to the described rotation by π/3.
[Section 3, counts of G2] The counts |V(G2)|=1066 and |E(G2)|=6264 are stated without derivation; they follow from the G1 counts, but a brief justification would help.
[Section 4, Table 2] The exhaustive search that found the six feasible labellings is not described; please state the search space and how the reduction rules are applied, and mention any software used and its version.
[Throughout] Please use a standard notation for the chromatic number (e.g., χ) consistently, and avoid undefined formatting artifacts.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the 5-chromatic graph is explicitly constructed and verified by independent SAT computations, with no fitted parameter renamed as a prediction.

full rationale

The paper's derivation chain is self-contained and non-circular. The graph H is defined by explicit trigonometric coordinates and Proposition 2.1 is proved by standard trig identities. The 84 unit vectors u_s are taken as the arcs of H, and the offset theta is derived from the geometry of H (the direction of -Q0P0), not fitted to force non-4-colorability. The graphs T5 and T6 are defined by polygonal paths of bounded length using these fixed vectors, and G0/G1 are obtained by a purely structural rule (membership in T5 or adjacency to at least 7 vertices of T5, then 7-core). The central claims—that (A,B) is non-monochromatic in G1, that G2 has the stated monochromatic-pair behavior, and that G3 is 5-chromatic—are verified by external SAT solvers on explicitly listed vertex sets and edge counts. Lemma 2.2 is an exhaustive numerical search, but it is an independent algebraic/circuit claim used to identify vertices, not a definition of the target result; any concern about floating-point rigor is a correctness risk, not circularity. Section 4 is explicitly self-described as unproved ('It would appear', 'presumably') and is not load-bearing for the main theorem. There are no self-citations, and no known result is renamed as a new framework. Therefore no circular step reduces the conclusion to its inputs.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The construction itself uses standard trigonometry and explicit finite vertex data, but the verification rests on two computational pillars that are not shipped: the floating-point exhaustive search in Lemma 2.2 and SAT solver runs. The epsilon thresholds and integer thresholds in the graph definition are hand-chosen parameters. Section 4 is explicitly incomplete and is not required for the main theorem. No new physical entities are posited.

free parameters (3)

epsilon thresholds in Lemma 2.2 = ε2=ε3=0.045, ε4=ε5=0.0067, ε6=0.0016, ε7=0.00089, ε8=0.00071
Hand-chosen numerical margins used in the exhaustive search to classify sums of unit vectors; the lemma's conclusion depends on them.
adjacency threshold in definition of G0 = 7
G0 keeps vertices adjacent to at least 7 vertices of V(T5); this integer is a hand-picked construction parameter that determines the 740-vertex G1.
path-length bounds for T5 and T6 = 5 and 6
These bounds control the size of the induced graphs; T5 alone does not make A and B non-monochromatic and T6 alone is too large, so both are used.

assumptions (5)

standard math Trigonometric identities (1) and (2) used in Proposition 2.1 are correct.
The cotangent telescoping and the inverse-sine sum identity from Fisher are standard; used to prove unit edge lengths in H.
ad hoc to paper The exhaustive double-precision search in Lemma 2.2 correctly classifies all sums of 2 to 8 unit vectors.
The proof is a stated computer verification, not a symbolic proof; the result is load-bearing for vertex counts of T5 and T6.
ad hoc to paper The SAT solver CaDiCaL correctly proves that G1 is not 4-colorable when A and B share a color.
No certificate is provided; the non-monochromatic pair claim rests on trusting the solver.
ad hoc to paper The Moser-spindle subgraph check on G3 was performed correctly.
The paper states it has been verified without giving a certificate or algorithm.
ad hoc to paper Section 4's assumption that feasible edge labellings extend to 4-colorings of the whole lattice.
The paper explicitly notes it has not proved this; this assumption is outside the main theorem.

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Cite this review

Pith. "Pith review of A Moser-spindle-free 5-chromatic unit distance graph on 2131 vertices in the plane." pith.science (2026). https://pith.science/paper/E3RNKHON

@misc{pith2026260804542,
  author       = {Pith},
  title        = {Pith review of: A Moser-spindle-free 5-chromatic unit distance graph on 2131 vertices in the plane},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E3RNKHON}},
  note         = {Machine review of arXiv:2608.04542}
}

read the original abstract

With regard to the Hadwiger-Nelson problem, several 5-chromatic unit distance graphs in the Euclidean plane have been discovered in recent years. While most constructions rely heavily on the Moser spindle, a few recent examples completely avoid it, the smallest one consisting of 1441 vertices. In this note, we introduce an original geometric approach to constructing such graphs by utilizing the arcs of a 7-fold symmetric unit distance graph on 21 vertices, and obtain a Moser-spindle-free 5-chromatic unit distance graph on 2131 vertices. While this is not a record small result, it arises from a straightforward, structured rule rather than a purely automated or brute-force search.

Figures

Figures reproduced from arXiv: 2608.04542 by the authors.

**Figure 2.** The unit distance graph H In view of (1) and (2), we have    α 2 + αβ + β 2 = α 2+β 2+(α+β) 2 2 = α 2+β 2+(−γ) 2 2 = 4 α 2 + αγ + γ 2 = α 2+γ 2+(α+γ) 2 2 = α 2+γ 2+(−β) 2 2 = 4 β 2 + βγ + γ 2 = β 2+γ 2+(β+γ) 2 2 = β 2+γ 2+(−α) 2 2 = 4 which is equivalent to    [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

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Reference graph

Works this paper leans on

10 extracted references · 4 canonical work pages

[1]

M. E. Fisher. Sum of inverse powers of cosines, SIAM Rev. 13 (1971), 116–119

1971
[2]

A. D. N. J. de Grey. The chromatic number of the plane is at least 5. Geombinatorics, vol. 28, no. 1, 2018, 18–31

work page 2018
[3]

M. J. H. Heule. Computing small unit-distance graphs with chromatic number 5. Geombi- natorics, vol. 28, no. 1, 2018, 32–50

work page 2018
[4]

M. J. H. Heule. Trimming graphs using clausal proof optimization. arXiv:1907.00929 [cs.LO] (2019). 6

arXiv 2019
[5]

M. J. H. Heule. Comments in Polymath16, thread 14, comment 24090, August 8, 2019

2019
[6]

M. J. H. Heule. Odd-Distance Virtual Edges in Unit-Distance Graphs. Geombinatorics, vol. 31, no. 2, 2021, 77-85

work page 2021
[7]

J. Parts. Graph minimization, focusing on the example of 5-chromatic unit-distance graphs in the plane. arXiv:2010.12665 [math.CO] (2020)

arXiv 2020
[8]

A. Soifer. The Mathematical Coloring Book, Springer, 2008, ISBN-13: 978-0387746401

2008

Show all 10 references

[9]

Voronov, A

V. Voronov, A. Neopryatnaya, E. Dergachev. Constructing 5-chromatic unit distance graphs embedded in the Euclidean plane and two-dimensional spheres. arXiv:2106.11824 [math.CO] (2021). Appendix A: The vertices ofG1 Below is a set of polygonal paths from (0,0) to (0, √

2021 arXiv
[10]

It was found using the Python library SetCoverPy

that collectively visit all vertices ofG 1, represented by their unit vector indices. It was found using the Python library SetCoverPy. 2 23 28 69 33 7 19 28 69 38 10 31 68 37 14 33 28 20 66 3 37 28 25 74 3 40 17 53 7 14 42 23 14 65 14 44 7 28 78 25 46 7 14 44 13 47 83 37 14 1...

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

M. E. Fisher. Sum of inverse powers of cosines, SIAM Rev. 13 (1971), 116–119

1971

[2] [2]

A. D. N. J. de Grey. The chromatic number of the plane is at least 5. Geombinatorics, vol. 28, no. 1, 2018, 18–31

work page 2018

[3] [3]

M. J. H. Heule. Computing small unit-distance graphs with chromatic number 5. Geombi- natorics, vol. 28, no. 1, 2018, 32–50

work page 2018

[4] [4]

M. J. H. Heule. Trimming graphs using clausal proof optimization. arXiv:1907.00929 [cs.LO] (2019). 6

arXiv 2019

[5] [5]

M. J. H. Heule. Comments in Polymath16, thread 14, comment 24090, August 8, 2019

2019

[6] [6]

M. J. H. Heule. Odd-Distance Virtual Edges in Unit-Distance Graphs. Geombinatorics, vol. 31, no. 2, 2021, 77-85

work page 2021

[7] [7]

J. Parts. Graph minimization, focusing on the example of 5-chromatic unit-distance graphs in the plane. arXiv:2010.12665 [math.CO] (2020)

arXiv 2020

[8] [8]

A. Soifer. The Mathematical Coloring Book, Springer, 2008, ISBN-13: 978-0387746401

2008