[#P3076] Borsuk’s conjecture in four dimensions

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 $i$?

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1Context

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.

2Problem setup

Definition 1 (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 2 (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 1. 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.

3What 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.

1Status

What counts as a solution

Current status (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.^[1][2]

1Packet records

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Record	Kind	Assessment
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.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 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][2] Relevance to this problem This is the dated publication status for the canonical target Borsuk’s conjecture in four dimensions. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details 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. Open full record →
The regular four-simplex gives b(4)≥5, while the current checked upper bound is b(4)≤8.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 2 The regular four-simplex gives b(4)≥5, while the current checked upper bound is b(4)≤8.^[1][2] Relevance to this problem Locates the present research frontier immediately below Borsuk’s conjecture in four dimensions. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details The regular four-simplex gives b(4)≥5, while the current checked upper bound is b(4)≤8. This leaves the following boundary unresolved: Determine whether b(4)=5; equivalently, prove five smaller-diameter parts always suffice or construct a unit-diameter set requiring at least six. The distinction is retained here so a restricted theorem, finite computation, or neighboring case is not presented as a solution of the full target. Related: Claim 1 Open full record →
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.	Trace	Evidence: Supported Backed by a cited source or by evidence short of a proof.
attempt · Route 1 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][2] Relevance to this problem Documents why Borsuk’s conjecture in four dimensions was treated as a distinct open target on 2026-08-01. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state completed Scope No narrower scope recorded Sources Reference 1, Reference 2 What happened The audit ran on 2026-08-01 across the cited primary literature and problem collections, Crossref, arXiv, and stable publisher records, TheoremDB published, prospecting, packet, formal, and retired corpus. It checked the exact statement, its named or normalized variants, recent status evidence, and the controlled TheoremDB corpus. Queries included: - Borsuk number R4 exact value - Borsuk conjecture dimension four five colors diameter graph - b(4) upper bound 8 Tolmachev Voronov 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. Related: Claim 1 Open full record →
Determine whether b(4)=5; equivalently, prove five smaller-diameter parts always suffice or construct a unit-diameter set requiring at least six.	Trace	Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 2 Determine whether b(4)=5; equivalently, prove five smaller-diameter parts always suffice or construct a unit-diameter set requiring at least six. Relevance to this problem Turns the remaining uncertainty in Borsuk’s conjecture in four dimensions into a checkable research target. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded What happened Research should address this boundary directly: Determine whether b(4)=5; equivalently, prove five smaller-diameter parts always suffice or construct a unit-diameter set requiring at least six. A claimed resolution should satisfy every item below and preserve the statement's exact quantifiers and normalizations: - 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. Related: Claim 1 Open full record →
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Notes and companion material

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.

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.

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ProblemBorsuk’s conjecture in four dimensions

All 3 recorded relations between these records and the problem

2See also

Tammes separation for fifteen points on the spheregeometry
Square peg problemgeometry
Birkhoff conjecture for integrable convex billiardsgeometry

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Plain text

“Borsuk’s conjecture in four dimensions.” TheoremDB. P3076. Problem statement; statement text SHA-256 8535f53c2dc7c7897ab81f0fb7064b3b7679b7315034ec4ac35d613f374b6ae5. https://theoremdb.org/statement/?ref=P3076

BibTeX

@misc{theoremdb-problem-8535f53c2dc7c7897ab81f0fb7064b3b7679b7315034ec4ac35d613f374b6ae5,
  title = {{Borsuk’s conjecture in four dimensions}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 8535f53c2dc7c7897ab81f0fb7064b3b7679b7315034ec4ac35d613f374b6ae5},
  url = {https://theoremdb.org/statement/?ref=P3076}
}

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This problem includes 4 records joined by 3 typed links, sourced from doi.org^[1], current as of August 1, 2026.

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1References

Packet source. 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. ↗preprint · primary source · arXiv:2605.19068, checked 2026-08-01 · checked 2026-08-01Source use: original summary.Proves the current upper bound b(4)≤8.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.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.
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. ↗preprint · primary source · arXiv:2511.03668v2, checked 2026-08-01 · checked 2026-08-01Source 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.

Original TheoremDB editorial statement and source synthesis; external works are used for citation only.

[#P3076] Borsuk’s conjecture in four dimensions

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