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

- ID: `P2876`
- Reference: `euclidean-three-space-rainbow-free-rhombus-colouring`
- Page: https://theoremdb.org/statements/P2876
- Record maturity: Reviewed problem with recorded work

## 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?

### Problem setup

- **Definition.** A rainbow triangle is a triangle whose three vertices receive three different colours.
- **Remark.** 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.

### What 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.

## 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. 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](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

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.

- Original intake status: 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.
- 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: The four prescribed points are cyclically adjacent at unit distance because alpha^2+beta^2=1, while AC and BD are the two diagonals.

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `euclidean-three-space-rainbow-free-rhombus-colouring`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>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. https://mathoverflow.net/questions/202861/3-colorings-of-the-unit-distance-graph-of-bbb-r3
   - Also cited at See dataset.references[0] for the exact external source and locator.
   - Also cited at Editorial research route recorded 2026-07-31
   - forum; reference source; checked 2026-07-31
   - Source use: citation_only
   - 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.
2. <a id="reference-2"></a>Alexey Glazyrin and Igor Pak, “Domes over curves”. DOI 10.1093/imrn/rnab138. arXiv:2005.02555 (2020). Full preprint relevant to A rainbow-free three-colouring of unit triangles in three-space. https://arxiv.org/abs/2005.02555
   - preprint; reference source; arXiv:2005.02555, checked 2026-07-31; checked 2026-07-31
   - Source use: citation_only
   - 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.
3. <a id="reference-3"></a>Mohammad Ghomi, “Equilaterally Triangulated Surfaces with Prescribed Boundary,” MathOverflow question 283921, asked October 20, 2017, checked 2026-08-01. question 283921 and its link sidebar, checked 2026-08-01 https://mathoverflow.net/questions/283921
   - forum; reference source; checked 2026-07-31
   - Source use: citation_only
   - 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.