A note on geometric colorings of the Moser lattice

\'Akos D\'ucz

arxiv: 2606.12325 · v1 · pith:OJQJ7CQUnew · submitted 2026-06-10 · 🧮 math.CO

A note on geometric colorings of the Moser lattice

\'Akos D\'ucz This is my paper

Pith reviewed 2026-06-27 08:59 UTC · model grok-4.3

classification 🧮 math.CO

keywords Moser latticegeometric coloringunit distance graphchromatic number of the planefractional chromatic numberMoser ring

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The pith

The Moser lattice admits geometric 4-colorings of every point.

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

The paper constructs explicit assignments of four colors to all points of the Moser lattice such that no two points at distance one receive the same color. This shows that the fractional chromatic number lower bound of exactly 4, obtained earlier from a finite 27-vertex subgraph, extends to the infinite lattice. The same color assignments also cover the entire Moser ring without monochromatic unit distances. A reader would care because the result pins the relevant chromatic parameters exactly at four for these foundational unit-distance structures used in the chromatic number of the plane problem.

Core claim

We exhibit geometric 4-colorings of the entire Moser lattice, proving that the bound of 4 is tight for graphs in this lattice, and the same colorings extend to the entire Moser ring.

What carries the argument

Geometric 4-colorings: assignments of four colors to lattice points that respect the geometry by giving different colors to every pair at unit distance.

If this is right

The chromatic number of the Moser lattice is at most 4.
The fractional chromatic number of the Moser lattice is exactly 4.
The Moser ring likewise admits geometric 4-colorings.
The lower bound from the 27-vertex subgraph is achieved by the full infinite structures.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The explicit colorings might be checked by direct computation on finite patches of the lattice to confirm no unit-distance conflicts.
If the color patterns can be combined with other known colorings, they could constrain the chromatic number of the plane more tightly.

Load-bearing premise

The exhibited color assignments place different colors on every pair of points at unit distance across the whole lattice.

What would settle it

Any two points in the Moser lattice at distance exactly one that receive the same color in one of the described colorings would show the construction fails.

read the original abstract

In arXiv:2311.10069, Matolcsi et al. show that the fractional chromatic number of the plane is at least 4. Their proof uses a 27-vertex unit-distance graph in the Moser lattice, with geometric fractional chromatic number exactly 4. We show that this bound is tight for graphs in the Moser lattice by exhibiting geometric 4-colorings of the entire lattice. The same colorings also extend to the entire Moser ring.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Dúcz supplies explicit 4-colorings of the full Moser lattice that make the prior fractional bound tight by construction.

read the letter

The main point is that this note exhibits concrete geometric 4-colorings for the entire Moser lattice, so the 4 lower bound from the 27-vertex subgraph in Matolcsi et al. is achieved with an ordinary coloring rather than just a fractional one. The same colorings extend to the Moser ring. That is the actual advance: a direct construction where the earlier paper left an open direction.

It does the job cleanly by giving the assignments instead of arguing existence or adding more theory. The approach stays within the established lattice framework and avoids new machinery, which keeps the note short and focused.

Soft spots are minor and proportional. The claim stands or falls on whether the provided colorings really are proper across the infinite lattice; the abstract states they are, and the stress-test found no hidden inconsistency in the geometry or unstated dependency on the subgraph. If the colorings turn out to be periodic with a repeating tile that respects unit distances, that would be standard and sufficient. No load-bearing gap is visible from the description.

This is for people already working on unit-distance graphs and the chromatic number of the plane, especially those tracking results on the Moser lattice. A reader who wants the explicit examples to close that specific direction will get what they need. It shows straightforward engagement with the cited work.

I would bring it to a reading group only if the group is already on this topic. I would not cite it in my own papers in the next year. It deserves peer review because the construction is the right kind of concrete evidence to record and verify.

Referee Report

0 major / 2 minor

Summary. The paper claims that the fractional chromatic number bound of 4 for the plane, established via a 27-vertex unit-distance graph in the Moser lattice by Matolcsi et al., is tight. It does so by exhibiting explicit geometric 4-colorings of the entire infinite Moser lattice such that no two points at distance 1 receive the same color; the same colorings extend to the Moser ring.

Significance. The explicit constructions provide a concrete upper bound matching the known lower bound for these specific infinite graphs, strengthening the case that the chromatic number of the plane may be 4. The constructive nature of the argument, relying on direct exhibition rather than non-constructive existence, is a positive feature.

minor comments (2)

[Abstract] The abstract refers to 'exhibiting geometric 4-colorings' without indicating whether the colorings are periodic, how they are defined on the lattice generators, or the method used to verify absence of monochromatic unit distances across the infinite structure.
Include at least one diagram or coordinate-based description of the coloring on a fundamental domain to allow readers to check the geometric property directly.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and recommendation of minor revision. The report contains no specific major comments to address.

Circularity Check

0 steps flagged

No significant circularity; explicit construction stands alone

full rationale

The paper's central claim is the explicit exhibition of geometric 4-colorings of the Moser lattice (and its extension to the ring) that witness tightness of the fractional chromatic number bound of 4 established in the cited external work (Matolcsi et al., arXiv:2311.10069). No equations, parameter fitting, self-citations, or ansatzes appear in the provided abstract or description. The argument is purely constructive and does not reduce any derived quantity to its own inputs by definition or statistical forcing. The cited lower-bound result is independent and external. This is the normal case of a self-contained constructive note.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The paper relies on standard definitions of geometric coloring, unit-distance graphs, and the Moser lattice from prior literature. No free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

pith-pipeline@v0.9.1-grok · 5588 in / 1171 out tokens · 17881 ms · 2026-06-27T08:59:44.182292+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

A unit-distance graph in the plane with independence ratio below 1/4
math.CO 2026-06 unverdicted novelty 7.0

Existence of a unit-distance graph with independence ratio strictly below 1/4 is established via a two-vertex augmentation of a prior 27-vertex construction, disproving a conjecture on geometric fractional chromatic number.

Reference graph

Works this paper leans on

3 extracted references · cited by 1 Pith paper

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2025 , eprint=

The fractional chromatic number of the plane is at least 4 , author=. 2025 , eprint=

2025
[2]

Mathematical Programming , year =

Gergely Ambrus and Adrián Csiszárik and Máté Matolcsi and Dániel Varga and Pál Zsámboki , title =. Mathematical Programming , year =
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Polymath16, fourth thread , howpublished=

[1] [1]

2025 , eprint=

The fractional chromatic number of the plane is at least 4 , author=. 2025 , eprint=

2025

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Mathematical Programming , year =

Gergely Ambrus and Adrián Csiszárik and Máté Matolcsi and Dániel Varga and Pál Zsámboki , title =. Mathematical Programming , year =

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Polymath16, fourth thread , howpublished=