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arxiv: 2605.19068 · v1 · pith:SSB2UN5Bnew · submitted 2026-05-18 · 🧮 math.MG

Reducing the upper bound for the Borsuk number in mathbb{R}⁴ to 8

Alexander Tolmachev , Vsevolod Voronov This is my paper

Pith reviewed 2026-05-20 08:08 UTC · model grok-4.3

classification 🧮 math.MG
keywords Borsuk numberb(4)unit diameterpartitiontruncated Lassak coverR^4discrete geometryupper bound
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The pith

Any set of unit diameter in four-dimensional Euclidean space can be partitioned into eight subsets of strictly smaller diameter.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper reduces the known upper bound on the Borsuk number b(4) from 9 to 8. It does this by constructing explicit partitions of several variants of the truncated Lassak cover, each split into eight parts whose diameters are all strictly less than 1. A sympathetic reader would care because the exact value of b(4) has remained open for decades and this step narrows the gap between the lower bound of 6 and the previous upper bound of 9. The result applies directly to the configurations previously handled by Lassak's 1982 construction.

Core claim

The Borsuk number b(4) satisfies b(4) ≤ 8. This is established by partitioning several variants of the truncated Lassak cover into 8 parts of diameter less than 1.

What carries the argument

Variants of the truncated Lassak cover partitioned into eight sets of diameter less than 1.

If this is right

  • Any unit-diameter set in R^4 can be partitioned into at most eight subsets of smaller diameter.
  • The 1982 Lassak upper bound of 9 is improved for the relevant truncated covers.
  • The minimal number of pieces sufficient to reduce diameter in four dimensions is at most 8.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same truncation and partitioning technique could be tested on candidate worst-case sets in five dimensions to check for similar reductions.
  • Computational enumeration of diameter-1 sets up to isometry could independently verify whether any evade the eight-part partitions.
  • The explicit partitions supply concrete examples that may help close the remaining gap to the known lower bound of 6.

Load-bearing premise

The variants of the truncated Lassak cover include all worst-case configurations that would require more than eight parts.

What would settle it

A concrete unit-diameter set in R^4 that cannot be partitioned into eight or fewer subsets each of strictly smaller diameter.

Figures

Figures reproduced from arXiv: 2605.19068 by Alexander Tolmachev, Vsevolod Voronov.

Figure 1
Figure 1. Figure 1: Truncated rhombic dodecahedron. form a universal covering system in the hyperplane p0. Moreover, for any set A′ of unit diameter containing the origin, there exists a rotation φ of p0 such that φA′ is covered by at least one of the sets Ui . Proof. Fix some set A of unit diameter. By Proposition 2, one can assume that its projection A′ contains the origin. By Proposition 3, we choose a pair of orthogonal v… view at source ↗
Figure 2
Figure 2. Figure 2: Lower bound, the Lassak cover ( 2a) and its truncate [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Circumscribed polyhedron approximation of Lassa [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
read the original abstract

The Borsuk number $b(n)$ of $n$-dimensional Euclidean space $\mathbb{R}^n$ is the smallest integer such that any set $F \subset \mathbb{R}^n$ of unit diameter can be partitioned into $b(n)$ subsets of strictly smaller diameter. For $n=4$, the best known upper bound $b(4) \leq 9$ follows from a construction by M. Lassak (1982). In the present paper, we construct partitions of several variants of the truncated Lassak cover into 8 parts of diameter less than 1, thereby showing that $b(4) \leq 8$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims to improve the upper bound on the Borsuk number b(4) from 9 to 8. It does so by constructing explicit partitions of several variants of the truncated Lassak cover (the configuration underlying Lassak's 1982 bound) into 8 subsets each of diameter strictly less than 1.

Significance. If the partitions are correct and the variants are shown to be exhaustive for the extremal cases, the result would be a concrete improvement on a long-standing bound in the Borsuk problem for dimension 4. The explicit, construction-based approach is a strength; machine-checked verification or reproducible coordinate lists for the partitions would further strengthen it.

major comments (2)
  1. [Abstract and §1] Abstract and §1: The claim that the 8-partitions of the truncated Lassak cover variants imply b(4) ≤ 8 requires an explicit reduction showing that every unit-diameter set F ⊂ R^4 is either contained in one of the treated variants or can be mapped to such a variant without increasing the minimal number of smaller-diameter parts. No such lemma or reference to a prior reduction is supplied, leaving the quantification over all F unaddressed.
  2. [Construction sections (e.g., §3–§5)] The manuscript focuses on the geometric construction for the specific covers but supplies no verification that these variants include all configurations that previously forced nine parts under Lassak's method. Without this, the improvement from 9 to 8 remains conditional on an unstated completeness assumption.
minor comments (2)
  1. Notation for the truncated Lassak cover and its variants should be introduced with a single diagram or coordinate table early in the paper to aid readability.
  2. Diameter calculations for the eight parts would benefit from an explicit table listing the maximum pairwise distances in each part.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful and detailed report. The major comments correctly identify that the manuscript does not explicitly articulate the reduction from an arbitrary unit-diameter set F to one of the treated variants of the truncated Lassak cover. We agree this step must be made rigorous and will revise the paper to supply it. We respond to each comment below.

read point-by-point responses

  1. Referee: [Abstract and §1] Abstract and §1: The claim that the 8-partitions of the truncated Lassak cover variants imply b(4) ≤ 8 requires an explicit reduction showing that every unit-diameter set F ⊂ R^4 is either contained in one of the treated variants or can be mapped to such a variant without increasing the minimal number of smaller-diameter parts. No such lemma or reference to a prior reduction is supplied, leaving the quantification over all F unaddressed.

    Authors: We acknowledge the gap. In the revised manuscript we will insert a new lemma (placed after the statement of the main result) that supplies the required reduction. The lemma will state that any set F of unit diameter in R^4 is congruent to a subset of one of the enumerated variants of the truncated Lassak cover, or can be affinely mapped into such a variant while preserving or decreasing all pairwise distances. The argument relies on the extremal properties already established by Lassak together with a case analysis on the possible supporting hyperplanes and diameter-realizing pairs; we will include a short proof sketch and a reference to the relevant parts of Lassak’s 1982 construction. revision: yes

  2. Referee: [Construction sections (e.g., §3–§5)] The manuscript focuses on the geometric construction for the specific covers but supplies no verification that these variants include all configurations that previously forced nine parts under Lassak's method. Without this, the improvement from 9 to 8 remains conditional on an unstated completeness assumption.

    Authors: We agree that an explicit completeness argument is missing. In the revision we will add a short subsection (new §2.3) that verifies the chosen variants are exhaustive for the configurations that required nine parts in Lassak’s original partition. The argument proceeds by enumerating the possible ways a 9-partition can arise from the truncated cover (via the positions of the four truncated vertices and the equatorial belt) and showing that each such configuration is isometric to one of the variants we partition into eight sets. This will remove the conditional character of the bound. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit geometric partitions of specific covers

full rationale

The paper's derivation consists of constructing explicit 8-partitions for several variants of the truncated Lassak cover, each with diameter less than 1. This is a direct constructive argument that does not invoke fitted parameters renamed as predictions, self-definitional loops, or load-bearing self-citations whose content reduces to the present work. The central claim follows from the geometry of the chosen covers without any reduction by construction to the input assumptions; the derivation remains self-contained against external benchmarks such as the prior Lassak bound.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the geometric validity of the constructed partitions and the assumption that the chosen variants of the Lassak cover are representative of all unit-diameter sets in R^4.

axioms (1)
  • standard math Euclidean distance defines diameter in the usual way
    Invoked in the definition of unit-diameter sets and smaller-diameter subsets.

pith-pipeline@v0.9.0 · 5641 in / 1032 out tokens · 45872 ms · 2026-05-20T08:08:59.940903+00:00 · methodology

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Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages · 2 internal anchors

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    work page