[#P2876] A rainbow-free three-colouring of unit triangles in three-space

A flat mathematical diagram showing unit triangles on rhombus faces colored with three colors. — A schematic view of unit triangles on rhombus faces colored with three colors.

Problem. Let \(\alpha,\beta>0\) satisfy \(\alpha^2+\beta^2=1\), with \(\alpha\) transcendental, and set \(A=(\alpha,0,0)\), \(B=(0,\beta,0)\), \(C=(-\alpha,0,0)\), and \(D=(0,-\beta,0)\). Does there exist a colouring \(c:\mathbb R^3\to\{1,2,3\}\) such that \(c(A)=c(B)=1\), \(c(C)=2\), \(c(D)=3\), and every equilateral triangle of side length \(1\) in \(\mathbb R^3\) has vertices of at most two colours?

1Context

Finite forced-colouring gadgets can accumulate even when a global construction remains elusive. Recording minimal obstructions and colour-propagation rules would prevent repeated searches through the same configurations.

2Problem setup

Definition 1 (A rainbow triangle). A rainbow triangle is a triangle whose three vertices receive three different colours.

Definition 2 (The four specified points form a planar rhombus with all four side lengths equal to 1; no condition). The four specified points form a planar rhombus with all four side lengths equal to 1; no condition is imposed on colours of pairs at unit distance unless they belong to a unit equilateral triangle.

Remark 1. Finite forced-colouring gadgets can accumulate even when a global construction remains elusive. Recording minimal obstructions and colour-propagation rules would prevent repeated searches through the same configurations.

3What counts as a solution

Construct such a colouring and prove the unit-equilateral-triangle condition for every triple of points in R^3, or prove that every colouring with the four prescribed values contains a rainbow unit equilateral triangle.
A finite obstruction proof may give a finite unit-distance configuration containing A,B,C,D and a complete certificate that every extension of the prescribed colours creates a rainbow triangle.

1Status

Current status (Current status and unresolved remainder). UNKNOWN as of 2026-07-31. The source page has zero answers and explicitly says its later application was largely resolved while this one-directional colouring problem remained open. No subsequent exact resolution was located. Construct such a colouring and prove the unit-equilateral-triangle condition for every triple of points in R^3, or prove that every colouring with the four prescribed values contains a rainbow unit equilateral triangle.^[1]

1Packet records

Record	Kind	Assessment
	Result	Supported
claim · Claim 1 UNKNOWN as of 2026-07-31. The source page has zero answers and explicitly says its later application was largely resolved while this one-directional colouring problem remained open. No subsequent exact resolution was located. Construct such a colouring and prove the unit-equilateral-triangle condition for every triple of points in R^3, or prove that every colouring with the four prescribed values contains a rainbow unit equilateral triangle.^[1] Relevance to this problem For euclidean three space rainbow free rhombus colouring, pins the dated research frontier: UNKNOWN as of 2026-07-31. The source page has zero answers and explicitly says its later application was largely resolved while this one-directional colouring problem remained open. No subsequent exact resolution was located. Construct such a colouring and prove the unit-equilateral-triangle. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1 Details UNKNOWN as of 2026-07-31. The source page has zero answers and explicitly says its later application was largely resolved while this one-directional colouring problem remained open. No subsequent exact resolution was located. A complete resolution must satisfy this condition: Construct such a colouring and prove the unit-equilateral-triangle condition for every triple of points in R^3, or prove that every colouring with the four prescribed values contains a rainbow unit equilateral triangle. Open full record →
	Trace	Reported
attempt · Route 1 Construct such a colouring and prove the unit-equilateral-triangle condition for every triple of points in R^3, or prove that every colouring with the four prescribed values contains a rainbow unit equilateral triangle.^[1] Relevance to this problem For A rainbow-free three-colouring of unit triangles in three-space, record euclidean-three-space-rainbow-free-rhombus-colouring-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Construct such a colouring and prove the unit-equilateral-triangle condition for every triple of points in R^3, or prove that every colouring with the four prescribed values contains a rainbow unit equilateral triangle. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Construct such a colouring and prove the unit-equilateral-triangle condition for every triple of points in R^3, or prove that every colouring with the four prescribed values contains a rainbow unit equilateral triangle. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Related: Claim 1 Open full record →
No records match these filters.

Notes and companion materialContext, examples, and computations

Original intake status. UNKNOWN as of 2026-07-31. The source page has zero answers and explicitly says its later application was largely resolved while this one-directional colouring problem remained open. No subsequent exact resolution was located.

On 2026-07-27 the Stack Exchange API reported zero answers, no accepted answer, and no closure for MathOverflow question 202861; all comments were read to disambiguate that the colouring need not be proper.
The 2020 paper arXiv:2005.02555 resolves much of the motivating configuration problem and explains the one-way connection in Section 5.4; the MathOverflow update states that this colouring extension remains open.
Searches for the prescribed transcendental rhombus, rainbow-free unit equilateral triangles, and three-colourings of Euclidean three-space found no later proof or obstruction.
A TheoremDB search for the coordinate quadruple, rainbow-free unit triangles, and the unit-distance graph formulation found no duplicate.

Recorded example 1. The four prescribed points are cyclically adjacent at unit distance because alpha^2+beta^2=1, while AC and BD are the two diagonals.

How the 2 records connectTyped relations and evidence flow

ProblemA rainbow-free three-colouring of unit triangles in three-space

Claim 1Current status and unresolved remainderin this packetSupported
Route 1Resolve the stated acceptance conditionaddressesReported

2See also

Conway’s thrackle conjecturediscrete geometry
Borsuk’s conjecture in four dimensionsdiscrete geometry
Completing a line arrangement to triangular bounded cellsdiscrete geometry

How to cite

TheoremDB contributors, “A rainbow-free three-colouring of unit triangles in three-space,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/euclidean-three-space-rainbow-free-rhombus-colouring

This page as plain text: euclidean-three-space-rainbow-free-rhombus-colouring.md

This problem includes 2 records joined by 1 typed links, sourced from mathoverflow.net^[1], current as of July 31, 2026.

1References

Packet source. MathOverflow: 3-colorings of the unit distance graph of R^3. Question 202861 and all visible comments, checked through the Stack Exchange API on 2026-07-27. Question 202861 and all visible comments, checked through the Stack Exchange API on 2026-07-27. ↗forum · discovery source · checked 2026-07-31Source use: original summary.UNKNOWN as of 2026-07-27. The source page has zero answers and explicitly says its later application was largely resolved while this one-directional colouring problem remained open. No subsequent exact resolution was located.Also cited at See dataset.references[0] for the exact external source and locator.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.For A rainbow-free three-colouring of unit triangles in three-space: UNKNOWN as of 2026-07-27. The source page has zero answers and explicitly says its later application was largely resolved while this one-directional colouring problem remained open. No subsequent exact resolution was located.Source named by the research packet.
MathOverflow: 3-colorings of the unit distance graph of R^3, source checked for the TheoremDB status review (2026-07-31). Status evidence identified in the source record and checked at the linked publication. ↗preprint · primary source · arXiv:2005.02555, checked 2026-07-31 · checked 2026-07-31Source use: original summary.UNKNOWN as of 2026-07-27. The source page has zero answers and explicitly says its later application was largely resolved while this one-directional colouring problem remained open. No subsequent exact resolution was located.Also cited at Full preprint relevant to A rainbow-free three-colouring of unit triangles in three-space.Source used to assess the problem's recorded status.For A rainbow-free three-colouring of unit triangles in three-space: UNKNOWN as of 2026-07-27. The source page has zero answers and explicitly says its later application was largely resolved while this one-directional colouring problem remained open. No subsequent exact resolution was located.
Mohammad Ghomi, “Equilaterally Triangulated Surfaces with Prescribed Boundary,” MathOverflow question 283921, asked October 20, 2017, checked 2026-08-01. Status evidence identified in the source record and checked at the linked publication. ↗forum · discovery source · checked 2026-07-31Source use: original summary.UNKNOWN as of 2026-07-27. The source page has zero answers and explicitly says its later application was largely resolved while this one-directional colouring problem remained open. No subsequent exact resolution was located.Also cited at question 283921 and its link sidebar, checked 2026-08-01.Source used to assess the problem's recorded status.For A rainbow-free three-colouring of unit triangles in three-space, this source asks a different equilateral-triangulation problem and was excluded as status evidence for the rainbow-free colouring target.

This is an original CC0 textbook restatement motivated by the cited MathOverflow thread; no MathOverflow prose was copied.

[#P2876] A rainbow-free three-colouring of unit triangles in three-space

1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

Recent contributions

2See also

1Lean verification

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

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1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

Recent contributions

2See also

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance