Discrete and convex geometry

Borsuk problem in four dimensions

Collaboration beta

Five vertices of a regular four-dimensional simplex already require five colors. The open question is whether five smaller-diameter pieces always suffice. The source closes several special and canonical face regimes, but its correction leaves noncanonical singleton and edge directions, buffered coverage, receiver selection, and cross-source safety open.

b (4) = 5

Known results and sources

The regular four-simplex supplies the five-part lower-bound example; it does not settle whether five pieces always suffice.

Research problem

Exact mathematical statement

Let $b(4)$ be the least number of parts needed so that every bounded subset of $\mathbb R^4$ of diameter one can be partitioned into subsets of diameter strictly below one. Determine whether

b(4)=5.

The governing source explicitly states that it does not contain a complete proof. Its dated external-state snapshot records $5\le b(4)\le 8$ .

Problem infographic

Problem at a glance

A five-vertex simplex, an unclosed colored convex body, and a finite graph with an open coral boundary share a dark geometric landscape. — The problem asks for a universal five-part partition; the source reports strong special regimes and a finite obstruction, while the middle geometry remains open.

Current mathematical picture

Where work on Borsuk problem in four dimensions stands

Open problem

The cumulative v7 source reports sharp local tetrahedral, capacity, canonical face-mode, and receiver-interface advances while correcting an overbroad canonical-scope claim, leaving the noncanonical BP4-T39 exclusion and the final contact and critical-core closures open.

Strongest supported footholdDirected-interval high-radius endpoint

For a compact unit-diameter set with squared circumradius $r>3/8$ , hence a support-minimal five-contact representation unless the already solved four-contact branch applies, $r\ge303/800$ implies a five-part partition; the geometric reduction is analytic and the final scalar box is supported by two directed floating-point interval exhaustions rather than exact arithmetic.

Evidence posture · Reported result

Leading routeSelf-consistent pole rounding

Route R3 — Self-consistent pole rounding Exploit simultaneously: $e_v\cdot e_w<-\frac14\quad(vw\in E), \tag{9.6}$ $\rho_v e_v\cdot(z-v) \ge\frac{|z-v|^2-\varepsilon}{2}, \tag{9.7}$ $|e_v-e_z|\ge|v-z|-\frac{\varepsilon}{|v-z|}, \tag{9.8}$ and the near-zero convex balance of at most five poles. The failure of ordinary nearest-simplex rounding is known, so the rounding cells must depend on the point configuration or the pole support data.

Route status · Active route

Useful failureCoarse cover-region graph coloring

Dead by BP4-D01. Use endpoint-sensitive geometry, not a region interaction graph.

Route status · Eliminated route

Main reductionFinite $K_4$ -free critical threshold core

A counterexample produces a finite induced six-critical fine-threshold graph whose coarse threshold supergraph is K4-free.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeNoncanonical face control and buffered complementary-2+2 coverage

Control arbitrary singleton and edge support directions through BP4-T39, keep canonical claims within their certified scope, and prove closed buffered cap coverage for the remaining complementary-2+2 modes.

Task status · Prerequisites still open

Later mathematical updatev7 sharpens the contact route and corrects its canonical scope

The cumulative v7 source adds BP4-T25–T39, records a receiver-axis dead route, and makes noncanonical face control the controlling open contact obligation.

v7 source revision order; not occurrence time or public priority

Work mapped so far

Borsuk problem in four dimensions in numbers

2.8kretained lines of mathematical investigation1,399 in the current working snapshot

Argument development: 2,411 · 85%
Explored or eliminated routes: 99 · 3%
Computational analysis: 98 · 3%
Open obligations: 129 · 5%
Definitions and setup: 112 · 4%

46inventoried working statements9routes investigated2reported milestones5open questions4contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Status checkedAug 26, 2026

Source recordedAug 30, 2026

Recommended next task

Convert the residual critical core into a contradiction

Either round the self-consistent poles, select a circuit forcing dominance/ $K_4$ , or exclude the clustered exact-diameter immersion.

Suggested move: Either round the self-consistent poles, select a circuit forcing dominance/ $K_4$ , or exclude the clustered exact-diameter immersion.

What would count as progress

Retain an exact proof or counterexample for the stated subproblem.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

23 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Active routeSelf-consistent pole rounding

Route status · Active route

Active routeLifted-circuit selection

Route R4 — Lifted-circuit selection Use the $(n-6)$ -dimensional stress space to choose a circuit whose small side is aligned with a large common neighborhood. The circuit transfer identity should then force either: • a coarse threshold $K_4$ , contradicting BP4-T08; • a dominated nonadjacent pair, contradicting criticality; • or a degree reduction to the degree-five branch. The missing step is combinatorial selection, not circuit algebra.

Route status · Active route

Active routeRobust common-neighbor or clustered-immersion theorem

Route R5 — Robust common-neighbor / clustered-immersion theorem Stabilize BP4-T19 for the almost-spherical common-neighbor window after contracting $O(\sqrt{1-\ell})$ clusters, or directly prove that a constant-width body cannot support the clustered strong $K_6$ immersion in the coarse- $K_4$ -free branch. Do not claim intrinsic linking already provides this theorem. Shared internal vertices and the projective direction construction must be handled explicitly.

Route status · Active route

Active routeLower-contact spindles

Route R6 — Lower-contact spindles Complete the regular-triangle contact-star assignment from BP4-T24, then perturb it to a quantitative three-contact theorem. Couple this with the four-contact strip to attack the middle-radius contact tree below $3/8$ .

Route status · Active route

Active routeFormal and independent certification

Route R7 — Formal and independent certification • Reimplement BP4-T23 with MPFR/Arb or exact dyadic intervals. • Formalize the lobe boundary-reduction lemma and monotonicity in $H$ . • Formalize the inherited tetrahedral KKT reductions. • Pin exact source theorem numbers for completion and the spherical arc-crossing lemma. ---

Route status · Active route

Active routeNoncanonical control and buffered direction-sheet cap coverage

The v7 source makes exact noncanonical singleton/edge control the highest-priority contact route, followed by scope-correct canonical exclusions and closed buffered complementary-edge cap coverage.

Route status · Active route

Explored alternatives

Other routes

3 recorded

Eliminated routeCoarse cover-region graph coloring

Dead by BP4-D01. Use endpoint-sensitive geometry, not a region interaction graph.

Route status · Eliminated route

Eliminated routeShared-face absorption

Route R1 — Shared-face absorption for residual five-contact lobes Priority: highest contact-route priority. Use the exact identity $M(z)-|z|^2 =\sum_j\gamma_j\bigl(M(z)-z\cdot v_j\bigr) \tag{9.1}$ for $z=\sum\gamma_jv_j$ . If $S_\eta=\{j:M(z)-z\cdot v_j\le\eta\}, \tag{9.2}$ then the total coefficient mass outside $S_\eta$ is at most $\frac{M(z)-|z|^2}{\eta}, \tag{9.3}$ and $\operatorname{dist}\bigl(z,\operatorname{conv}\{v_j:j\in S_\eta\}\bigr) \le\frac{M(z)-|z|^2}{\eta}. \tag{9.4}$ Near-extremal rays therefore lie near proper faces shared with neighboring normal cells. The route must quantify the receiving cells' spare diameter margin and choose a globally consistent transfer rule.

Route status · Eliminated route

Narrowed routeAnalytic balanced-ray theorem

Route R2 — Analytic balanced-ray theorem Prove $F_{s,H}(P,Q) \le 4c(s) \left( \sqrt{\frac{1-s+c(s)}{H+c(s)}}-1 \right)^2. \tag{9.5}$ This would replace the $303/800$ interval tree by a one-variable proof and sharpen the audit boundary. It will not alone reach $3/8$ , so it is secondary to R1 unless pursued as proof engineering.

Route status · Narrowed route

Route statements and reductions

Statements the next route can inspect and build on

Route statementRank-six induced sign matrix and lifted circuits

For a finite induced threshold realization, the induced threshold sign matrix has rank at most six, is conditionally positive semidefinite, and supplies the stated lifted-circuit distance-transfer bounds.