# P3076: Borsuk’s conjecture in four dimensions

- ID: `P3076`
- Reference: `borsuk-conjecture-in-four-dimensions`
- Page: https://theoremdb.org/statements/P3076
- Record maturity: Reviewed problem with recorded work

## Problem

Does every bounded set \(S\subset\mathbb R^4\) of diameter \(1\) admit a partition \(S=S_1\cup\cdots\cup S_5\) with \(\operatorname{diam}(S_i)<1\) for every \(i\)?

### Context

Known frontier: The regular four-simplex gives b(4)≥5, while the current checked upper bound is b(4)≤8.

Open boundary: Determine whether b(4)=5; equivalently, prove five smaller-diameter parts always suffice or construct a unit-diameter set requiring at least six.

### Problem setup

- **Definition (Borsuk number b(4)).** The least m such that every unit-diameter subset of R⁴ can be partitioned into m sets of diameter strictly below one.
- **Definition (diameter graph).** For a finite set, the graph joining pairs at distance equal to the set diameter; a smaller-diameter partition is a proper coloring.
- **Remark.** Write b(4) for the smallest number of strictly smaller-diameter parts needed for every unit-diameter subset of four-dimensional Euclidean space. A regular four-simplex forces b(4)≥5. The question asks whether five parts always suffice.

### What counts as a solution

- For a proof, partition every bounded unit-diameter subset of R⁴ into five strictly smaller-diameter classes.
- For a disproof, give an explicit finite or compact unit-diameter set whose diameter graph needs at least six colors.

## Status

OPEN as checked on 2026-08-01. Strongest checked neighboring result: The regular four-simplex gives b(4)≥5, while the current checked upper bound is b(4)≤8. Exact unresolved remainder: Determine whether b(4)=5; equivalently, prove five smaller-diameter parts always suffice or construct a unit-diameter set requiring at least six. [1](#reference-1) [2](#reference-2)

## Work

### Evidence for the current status

**Claim 1 (Current status and exact unresolved remainder).** OPEN as checked on 2026-08-01. Strongest checked neighboring result: The regular four-simplex gives b(4)≥5, while the current checked upper bound is b(4)≤8. Exact unresolved remainder: Determine whether b(4)=5; equivalently, prove five smaller-diameter parts always suffice or construct a unit-diameter set requiring at least six.

The problem was checked as open on 2026-08-01.

The strongest neighboring result found in the cited sources is: The regular four-simplex gives b(4)≥5, while the current checked upper bound is b(4)≤8.

The exact unresolved remainder is: Determine whether b(4)=5; equivalently, prove five smaller-diameter parts always suffice or construct a unit-diameter set requiring at least six.

A complete resolution must meet the following acceptance conditions:
- For a proof, partition every bounded unit-diameter subset of R⁴ into five strictly smaller-diameter classes.
- For a disproof, give an explicit finite or compact unit-diameter set whose diameter graph needs at least six colors.

### Background and intake notes

- Original intake status: OPEN as checked on 2026-08-01. Strongest checked neighboring result: The regular four-simplex gives b(4)≥5, while the current checked upper bound is b(4)≤8. Exact unresolved remainder: Determine whether b(4)=5; equivalently, prove five smaller-diameter parts always suffice or construct a unit-diameter set requiring at least six.
- The release review checked 2 structured sources on 2026-08-01.
- Equivalent-formulation queries: Borsuk number R4 exact value; Borsuk conjecture dimension four five colors diameter graph; b(4) upper bound 8 Tolmachev Voronov
- Strongest checked neighboring result: The regular four-simplex gives b(4)≥5, while the current checked upper bound is b(4)≤8.
- Exact unresolved remainder: Determine whether b(4)=5; equivalently, prove five smaller-diameter parts always suffice or construct a unit-diameter set requiring at least six.

### Other known results

- **Claim 2** (supported): The regular four-simplex gives b(4)≥5, while the current checked upper bound is b(4)≤8. [1](#reference-1) [2](#reference-2)

### Prior approaches

- **Route 1** (supported): The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: The regular four-simplex gives b(4)≥5, while the current checked upper bound is b(4)≤8. Unresolved remainder: Determine whether b(4)=5; equivalently, prove five smaller-diameter parts always suffice or construct a unit-diameter set requiring at least six. [1](#reference-1) [2](#reference-2)

### Open directions

- **Route 2** (reported): Determine whether b(4)=5; equivalently, prove five smaller-diameter parts always suffice or construct a unit-diameter set requiring at least six.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `borsuk-conjecture-in-four-dimensions`, 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>Tolmachev, Alexander and Voronov, Vsevolod, “Reducing the upper bound for the Borsuk number in $\mathbb{R}^4$ to 8”. arXiv (2026). DOI 10.48550/arXiv.2605.19068. abstract and main constructions https://doi.org/10.48550/arXiv.2605.19068
   - Also cited at A. Tolmachev and V. Voronov, “Reducing the upper bound for the Borsuk number in R⁴ to 8,” arXiv:2605.19068 (2026). abstract and main constructions
   - preprint; primary source; arXiv:2605.19068, checked 2026-08-01; checked 2026-08-01
   - Source use: original_summary
   - Proves the current upper bound b(4)≤8.
   - Source used to assess the problem's recorded status.
   - For Borsuk’s conjecture in four dimensions: This is the dated publication status for the canonical target Borsuk’s conjecture in four dimensions.
   - Source named by the research packet.
2. <a id="reference-2"></a>O. R. Musin, “Borsuk’s conjecture for two-distance sets and its equivalent formulation for graphs,” arXiv:2511.03668v2 (2025). abstract and graph-equivalence section https://arxiv.org/abs/2511.03668v2
   - preprint; primary source; arXiv:2511.03668v2, checked 2026-08-01; checked 2026-08-01
   - Source use: original_summary
   - Records dimension four as open and develops the graph formulation for finite two-distance sets.
   - Source used to assess the problem's recorded status.
   - For Borsuk’s conjecture in four dimensions: Records dimension four as open and develops the graph formulation for finite two-distance sets.