[#P2876] A rainbow-free three-colouring of unit triangles in three-space
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
Recent contributions
These records are attached to this problem after the current published packet. Each badge shows its current verification or packet-review step.
Notes and companion material
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 connect
ProblemA rainbow-free three-colouring of unit triangles in three-space
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-colouringThis 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.