ProofAtlas

Extremal combinatorics · graph coloring · intersecting set systems

Stahl’s Multichromatic Kneser-Graph Conjecture

Collaboration beta

When every k-subset of [n] receives s colors and disjoint subsets receive disjoint palettes, Stahl’s conjecture gives the exact minimum total number of colors.

χs(KG(n,k))=qn-2r,s=qk-r
Known results and sources
An editorial Kneser-graph cover pairs disjoint k-subsets with nonoverlapping multicolor palettes, arranged around a restrained crown-and-star motif without implying a proof of Stahl’s conjecture.
Disjoint k-subsets must receive disjoint s-color palettes; this low-text cover identifies Stahl’s multichromatic Kneser-graph problem and is not evidence.

Research problem

Exact mathematical statement

Let KG(n,k)KG(n,k) be the graph whose vertices are the kk-subsets of [n][n], with two vertices adjacent exactly when the subsets are disjoint. Let χs(G)\chi_s(G) be the least number of colors in an ss-fold coloring, where every vertex receives an ss-element palette and adjacent vertices receive disjoint palettes. For n≥2kn\ge 2k, write s=qk-rs=qk-r with q≥1q\ge1 and 0≤r≤k-10\le r\le k-1. Stahl’s conjecture is

χs(KG(n,k))=qn-2r.\chi_s(KG(n,k))=qn-2r.

Equivalently,

χs(KG(n,k))=2s+(n-2k)⌈s/k⌉.\chi_s(KG(n,k))=2s+(n-2k)\lceil s/k\rceil.

Problem infographic

Problem at a glance

A problem-centered scientific plate for Stahl’s multichromatic Kneser-graph conjecture shows two disjoint k-subsets as adjacent Kneser vertices with separate s-color palettes, states the conjectured value χₛ(KG(n,k)) = qn − 2r for s = qk − r, and gives the exact boundary example KG(4,2) as six vertices paired by three matching edges with χₛ = 2s.
In KG(n,k), vertices are k-subsets of [n] and adjacency means disjointness. An s-fold coloring assigns s colors to every vertex and disjoint palettes to adjacent vertices. Stahl’s conjecture predicts χₛ(KG(n,k)) = qn − 2r when s = qk − r. The KG(4,2) matching is an exact boundary example; the general conjecture remains open.

Current mathematical picture

Where work on Stahl’s Multichromatic Kneser-Graph Conjecture stands

Open conjecture

The current work gives two exact reformulations of Stahl’s conjecture, derives exact residual-design identities and local constraints, proves the shadow-localization range and all cases with k at most 3, reproduces two finite trace calculations, and reduces the first unresolved residual layer to a dense five-color partition problem. It also records sharp limits of scalar, coordinatewise, single-crown, topological, and unrestricted peeling routes. The general conjecture remains open, and the separate (11,4,45) result remains an uncertified candidate.

Strongest supported footholdExact residual invariants and local hierarchy

Exact moments, fixed deficit, and the minimal-counterexample transversal hierarchy become simultaneous constraints.

Evidence posture · Reported result
Leading routeDense partition-aware shadow

Exploit the five-way partition, completeness, monochromatic face, middle band, and codimension-two link budget simultaneously.

Route status · Active route
Useful failureUnrestricted exact-layer peeling

Refuted by a nine-color 3-fold KG(5,2) example; threshold-specific or local-layer variants remain possible.

Route status · Refuted route
Main reductionLC₁ and exact Z=2 systems

One-extra-color localization is proved, and the concentrated and split Z=2 cases are expressed with their exact common shadow-and-slack budget.

Evidence posture · Reported reduction
Completed special caseShadow range and k≤3 completed

The current work derives Stahl’s formula for r(n−2k)≤k and, separately, for every multiplicity with k≤3.

Evidence posture · Reported special case
Evidence footholdTrace-capacity exclusions for (12,5)

The reproduced trace-capacity calculation records 80 eliminated high-defect profiles for (n,k)=(12,5), including Z=9 profiles (9) and (8,1); the separate diversity-shadow argument eliminates Z=11.

Evidence posture · Computation reproduced · provisional · dependencies incomplete
Priority open bridgeProve the dense partition-aware shadow theorem

Rule out every complete five-class partition of the k-subsets of [2k+1] satisfying the monochromatic-face, middle-band, link-shadow, completeness, and transversal constraints.

Task status · Ready to work on
Research-record correctionResearch-record correction

We corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Stahl’s Multichromatic Kneser-Graph Conjecture in numbers

1.7kretained lines of mathematical investigation1,690 in the current working snapshot
Argument development
1,434 · 85%
Explored or eliminated routes
63 · 4%
Computational analysis
35 · 2%
Open obligations
41 · 2%
Definitions and setup
117 · 7%
19selected mapped statements16routes investigated9reported milestones7open questions6contribution-ready tasks
Evidence attached to the current work2 original computation reruns
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.

Proposed bySaul StahlProposed1976Status checkedAug 2, 2026
Source recordedAug 2, 2026

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

27 selected steps

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

27 selected steps

Scroll horizontally to explore the route

Working route overview for Stahl’s Multichromatic Kneser-Graph ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Candidate χ₄₅(KG(11,4))=126 — ChallengedCandidate χ₄₅(KG(11,4))=126Stahl’s multichromatic Kneser-graph conjecture — Depends on missing premiseStahl’s multichromaticKneser-graph conjectureConcentrated defect reduces to ordinary localization — Depends on missing premiseConcentrated defect reducesto ordinary localizationDense LC₂ five-color normal form — Depends on missing premiseDense LC₂ five-color normalformExact residual-design equivalence — Depends on missing premiseExact residual-designequivalenceThreshold-localization equivalence — Depends on missing premiseThreshold-localizationequivalenceWeighted Kneser blow-up formulation — Depends on missing premiseWeighted Kneser blow-upformulationAggregate diversity lower bound — Depends on missing premiseAggregate diversity lowerboundAll cases with k at most 3 — Depends on missing premiseAll cases with k at most 3At least d+1 residual-list maxima — Depends on missing premiseAt least d+1 residual-listmaximaExact residual moment hierarchy — Depends on missing premiseExact residual momenthierarchyExact weighted Chen-crown capacity — Depends on missing premiseExact weighted Chen-crowncapacityDense partition-aware shadow — activeDense partition-aware shadowBlocker complexity versus shadow cost — activeBlocker complexity versusshadow costCoupled trace-vector stability — activeCoupled trace-vectorstabilityCapacitated Hilton–Milner packing — activeCapacitated Hilton–MilnerpackingScalar Hilton–Milner deficit counting — stoppedScalar Hilton–Milner deficitcountingIndependent Kruskal–Katona optimization of each trace column — stoppedIndependent Kruskal–Katonaoptimization of each tracecolumnPromoting residual-list maximality to global maximality — stoppedPromoting residual-listmaximality to globalmaximalityA single weighted Chen crown — stoppedA single weighted Chen crownProve the dense partition-aware shadow theorem — OpenProve the densepartition-aware shadowtheoremRelate blocker complexity to shadow cost — OpenRelate blocker complexity toshadow costDevelop coupled trace-vector stability — OpenDevelop coupled trace-vectorstabilityClose capacitated Hilton–Milner packing — OpenClose capacitatedHilton–Milner packingProve multi-crown incompatibility — OpenProve multi-crownincompatibilityProve a weighted localization bridge — OpenProve a weightedlocalization bridgeComplete the independent (11,4,45) verifier — BlockedComplete the independent(11,4,45) verifier
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 routeDense partition-aware shadow

Exploit the five-way partition, completeness, monochromatic face, middle band, and codimension-two link budget simultaneously.

Route status · Active route
Active routeBlocker complexity versus shadow cost

Charge the four required units of blocker chromatic excess against the exact Z=2 shadow-and-saturation-slack budget.

Route status · Active route
Active routeCoupled trace-vector stability

Replace independent point-column optimization with whole-row trace-vector structure near Hilton–Milner equality.

Route status · Active route
Active routeCapacitated Hilton–Milner packing

Coordinate center–exception pairs, exact moments, and higher-rank transversal capacities for the low-diversity branch.

Route status · Active route
Active routeHereditary weighted localization or weighted Schrijver

A point-increment localization theorem would induct on defect weight, while weighted Schrijver would prove the residual lower bound in one step.

Route status · Active route

Explored alternatives

Other routes

11 recorded
Narrowed routeMulti-crown incompatibility

The selector branch survives only through several crowns or another configuration whose exact defect-slot colors cannot be globally reused.

Route status · Narrowed route
Narrowed routeSelector-free palette covering

Use the fibre-palette covering and leakage laws; concentrated defect is routed to localization, while distributed defect still needs diversity and leakage coupling.

Route status · Narrowed route
Route held in reserveIndependent (11,4,45) verifier

Recover the missing definitions, correct the checker, emit certificates, and independently reimplement the finite candidate before treating it as established.

Route status · Route held in reserve
Browse 8 more explored routes
Useful but insufficientScalar Hilton–Milner alone

Useful for broad elimination and k≤3, but too weak as an independent-family bound in the first unresolved k≥4 cases.

Route status · Useful but insufficient
Useful but insufficientIndependent trace-column optimization

Useful as a finite relaxation, but it ignores the common family rows required by a genuine trace matrix.

Route status · Useful but insufficient
Not yet justifiedExact triple transversality

The earlier σ=0 conclusion used an unjustified promotion from residual-list maximality to global maximality.

Route status · Not yet justified
Useful but insufficientSingle Chen crown

The exact perfect-graph calculation exposes genuine tail capacity, so one crown cannot close distributed defect.

Route status · Useful but insufficient
Refuted routeUnrestricted exact-layer peeling

Refuted by a nine-color 3-fold KG(5,2) example; threshold-specific or local-layer variants remain possible.

Route status · Refuted route
Useful but insufficientOrdinary topological lower bounds

Base Kneser-color forcing does not yield one distinct color per defect slot at one common core.

Route status · Useful but insufficient
Route held in reserveRaw residual-design MILP

The unsymmetrized formulation stalled under color symmetry; orbit-, trace-, blocker-, or palette-level symmetry breaking is the retained computational direction.

Route status · Route held in reserve
Eliminated routeNaive LC₁ iteration

Recolored vertices can split among several retained colors, so one removed color may create several localized colors.

Route status · Eliminated route

Route statements and reductions

Statements the next route can inspect and build on

Route statementWeighted Kneser blow-up formulation

An exact residual design is equivalent to a coloring of H_z with at most d+Z−1 colors; Stahl’s lower bound becomes χ(H_z)≥d+Z.

Source-reported route statement · dependencies incomplete
Route statementThreshold-localization equivalence

Stahl’s conjecture is equivalent to the threshold statement that every s-fold coloring of KG(m,k) has a (k−1)-set B whose disjoint-side palette union has size at least s+(m−2k+1)⌈s/k⌉.

Source-reported route statement · dependencies incomplete
Route statementRank-by-rank transversal hierarchy

In a minimal exact residual counterexample, every h-set X with 1≤h≤n−2k transverses at most h+z(X)−1 residual families.

Source-reported route statement · dependencies incomplete
Route statementAggregate diversity lower bound

Every exact residual design satisfies a lower bound on the sum of family diversities that separates balanced and nonbalanced center vectors and can force an extra deficit beyond formal Hilton–Milner equality.

Source-reported route statement · dependencies incomplete
Route statementTrace-capacity exclusions for (12,5)

The reproduced trace-capacity calculation records 80 eliminated high-defect profiles for (n,k)=(12,5), including Z=9 profiles (9) and (8,1); the separate diversity-shadow argument eliminates Z=11.

Computation reproduced · provisional · dependencies incomplete
Route statementExact Z=2 shadow-and-slack systems

The concentrated and split Z=2 trace systems have different pointwise multiplicities but the same exact total shadow-and-saturation-slack budget; in the concentrated case every core transverses at least three classes, not necessarily exactly three.

Source-reported route statement · dependencies incomplete
Route statementDense LC₂ five-color normal form

At m=2k+1, a hypothetical concentrated Z=2 obstruction yields a complete five-color partition with a monochromatic (k+1)-face, at least two middle-band-only classes, at most two colors in every (k−2)-link, and at least three transversal classes at every (k−1)-core.

Source-reported route statement · dependencies incomplete

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

7 featured tasks
01
Prove the dense partition-aware shadow theorem

Rule out every complete five-class partition of the k-subsets of [2k+1] satisfying the monochromatic-face, middle-band, link-shadow, completeness, and transversal constraints.

Suggested move: Double-count (k−2)-shadow multiplicities by intersection layer with the monochromatic face while keeping the partition and completeness data.
Ready to work on
02
Close capacitated Hilton–Milner packing

Show that the near-Hilton–Milner residual families required by the fixed deficit cannot fit all pair, center, exception, and higher-rank transversal capacities.

Suggested move: Combine center–exception pair packing with exact point moments and known Hilton–Milner trace vectors.
Ready to work on
03
Prove a weighted localization bridge

Establish weighted palette drop, hereditary star-localization, or weighted Schrijver in a form that certifies globally distinct defect-slot colors.

Suggested move: Start with the threshold local-layer increment or a point-increment theorem for H_z.
Ready to work on
04
Relate blocker complexity to shadow cost

Bound disjoint-shadow excess in terms of blocker chromatic excess strongly enough that four units of blocker excess exceed the exact shadow-and-slack budget.

Suggested move: Establish a sharp single-family or partition-level inequality for ε(𝓗) versus r(𝓗)−1 that retains saturation slack.
Ready to work on
05
Develop coupled trace-vector stability

Characterize or majorize whole family trace vectors near Hilton–Milner equality, then combine row equations, column equations, and pair capacities.

Suggested move: Classify the first several deficit levels by complete trace vector rather than optimize coordinates independently.
Ready to work on
06
Prove multi-crown incompatibility

Find several selector crowns whose remainder tails cannot hide or globally reuse all distributed defect units.

Suggested move: Treat the exact single-crown formula as the base constraint and coordinate multiple matched crowns.
Ready to work on
07
Complete the independent (11,4,45) verifier

Recover or redefine the equality families and tuple convention, correct the endpoint assertion, emit explicit certificates, and independently reimplement the checker.

Suggested move: Recover the original verifier and missing definitions before rerunning with assertions enabled and without -DNDEBUG.
Blocked by the current route

Sourced mathematical context

The known mathematical landscape

Context collected Aug 2, 2026
Current statusOpen conjecture

The conjectured formula remains open in general. The literature records complete proofs for several parameter families and additional residue ranges, but the 2024 notes continue to treat the general formula as a conjecture.

[1]
External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

  1. PreprintVan den Heuvel and Xu develop new results for multicolouring transfer questions and give simpler proofs of several known cases of Stahl's conjecture.[1]
  2. Peer reviewedOsztenyi proves the formula when 2n<m<3n and 0<=r<n/(m-2n) for tuple multiplicity nq-r.[3]
  3. Peer reviewedKincses, Makay, Maroti, Osztenyi, and Zadori prove the special case corresponding to m=10 and n=4 for all multiplicities.[4]
  4. Peer reviewedStahl's later work completes the conjecture for Kneser-graph set-size parameters n=2 and n=3 and for ground-set size m=2n+1.[3]
6 cited sources3 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusStahl's Multichromatic Kneser-Graph Conjecture
Solved special caseLovasz-Kneser theorem

The tuple-multiplicity-one case is the chromatic-number formula for Kneser graphs proved by Lovasz.

[3]
Equivalent formulationgraph homomorphisms between Kneser graphs

An s-tuple colouring with t colours is a graph homomorphism into a Kneser graph, so the conjecture can be phrased as a homomorphism nonexistence problem.

[1]
Logical consequenceasymptotic Poljak-Rodl product-colouring bound

A published result shows that Stahl's conjecture would imply an asymptotic upper bound of 1/2+epsilon for the normalized chromatic loss under categorical graph products.

[5]

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details
Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

How the route was assembled

Argument structure

These stages follow the mathematical order of the supplied argument.

10 mapped milestonesretained argument map

Browse all 10 mapped stages

  1. stage 1Exact statement and explicit upper bound
  2. stage 2Two exact full-conjecture reductions
  3. stage 3Exact residual invariants and transversals
  4. stage 4Canonical residual structure forces Z≥2
  5. stage 5Shadow range and k≤3 cases
  6. stage 6Trace-capacity calculations reproduced
  7. stage 7Single-crown route sharply limited
  8. stage 8LC₁ and exact Z=2 systems
  9. stage 9Dense LC₂ frontier isolated
  10. stage 10Finite (11,4,45) candidate held uncertified
Exact statement and explicit upper boundThe current work fixes Stahl’s formula and supplies the standard construction attaining qn−2r colors.

Mapped research milestoneInitial research sequence

Research stage 1
Two exact full-conjecture reductionsThe lower bound is made equivalent both to excluding exact residual designs and to proving threshold localization.

Mapped research milestoneInitial research sequence

Research stage 2
Exact residual invariants and transversalsPointwise exactness yields the complete moment hierarchy and, under minimality, rank-by-rank transversal capacities.

Mapped research milestoneInitial research sequence

Research stage 3
Canonical residual structure forces Z≥2Uncrossing and a Chen-crown incidence count force at least d+1 distinct residual-list maxima.

Mapped research milestoneInitial research sequence

Research stage 4
Shadow range and k≤3 casesThe current work proves the conjectured formula when r(n−2k)≤k and for every multiplicity with k≤3.

Mapped research milestoneInitial research sequence

Research stage 5
Trace-capacity calculations reproducedTwo mounted (12,5) scripts reproduce their logs byte-for-byte and retain 80 high-defect profile exclusions.

Mapped research milestoneInitial research sequence

Research stage 6
Single-crown route sharply limitedExact perfect-graph capacity shows that one Chen crown cannot charge the distributed defect tail.

Mapped research milestoneInitial research sequence

Research stage 7
LC₁ and exact Z=2 systemsOne-extra-color localization is proved and both Z=2 defect profiles are reduced to exact trace equations with retained saturation slack.

Mapped research milestoneInitial research sequence

Research stage 8
Dense LC₂ frontier isolatedThe immediate bridge becomes a five-class partition-aware shadow contradiction with explicit dense-boundary constraints.

Mapped research milestoneInitial research sequence

Research stage 9
Finite (11,4,45) candidate held uncertifiedA checker assertion failure and missing definitions and certificates prevent the retained finite argument from becoming an established result.

Mapped research milestoneInitial research sequence

Research stage 10

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

17 standing statements1 proposed statements1 challenged statements9 mathematical milestones7 open questions2 narrowed routes8 conditional results2 completed special cases
Statements by mathematical role19 selected mapped statements
  • theorem candidate2 of 192
  • lemma10 of 1910
  • equivalence3 of 193
  • computational claim1 of 191
  • negative result1 of 191
  • reduction2 of 192
Checks attached to these statementsPositive checks available
  • Original computation rerun2
Selected mathematical clusters8 mathematical clusters
Conjecture and exact reductionsThe exact statement, explicit upper construction, residual-design normalization, weighted blow-up, and threshold-localization equivalence.6 displayed rows · 1 route included
  • retained route statementStahl’s multichromatic Kneser-graph conjecture
  • retained route statementExplicit upper construction
  • retained route statementExact residual-design equivalence
  • retained route statementWeighted Kneser blow-up formulation
  • retained route statementThreshold-localization equivalence
  • Active routeHereditary weighted localization or weighted SchrijverA point-increment localization theorem would induct on defect weight, while weighted Schrijver would prove the residual lower bound in one step.
Exact residual structureExact moments, minimal-counterexample transversal bounds, and canonical residual-list maxima.6 displayed rows · 1 route included
  • retained route statementExact residual moment hierarchyconditional
  • retained route statementRank-by-rank transversal hierarchyconditional
  • retained route statementAt least d+1 residual-list maximaconditional
  • DerivationSum the exact pointwise residual equation over all k-sets containing a fixed subset R; specializing R gives total and point moments, and subtracting from pα fixes the total EKR deficit.active reported
  • DerivationUncross cross-intersecting residual pairs under the quadratic potential, assign vertices to hypothetical d maxima, and use Chen’s matched crown to contradict the available Z−1 subordinate slots.active reported
  • Active routeCapacitated Hilton–Milner packingCoordinate center–exception pairs, exact moments, and higher-rank transversal capacities for the low-diversity branch.
Source-reported solved rangesThe shadow-localization inequality and the completed k≤3 family, both still recorded as provisional narrative results.3 displayed rows · 1 route included
  • retained route statementShadow-localization solved range
  • retained route statementAll cases with k at most 3special case
  • Useful but insufficientScalar Hilton–Milner aloneUseful for broad elimination and k≤3, but too weak as an independent-family bound in the first unresolved k≥4 cases.
Diversity and trace programAggregate diversity, exact trace capacity, reproduced finite calculations, and the coupled-vector obligation.7 displayed rows · 2 routes included
  • retained route statementAggregate diversity lower boundconditional
  • retained route statementTrace-capacity exclusions for (12,5)computational
  • ComputationRerun of the mounted (12,5) trace-loss verification script under Python 3.13.5.The rerun exited successfully and reproduced the mounted log byte-for-byte; the loss-only relaxation contradicts concentrated Z=10 and Z=11 and the tested z_x=10, Z=11 case, but not concentrated Z=9. · reproduced same implementation
  • ComputationRerun of the mounted (12,5) trace-capacity verification script under Python 3.13.5.The rerun reproduced the mounted log byte-for-byte and records 80 eliminated high-defect partitions, including Z=9 profiles (9) and (8,1). · reproduced same implementation
  • Research targetDevelop coupled trace-vector stabilityopen
  • Active routeCoupled trace-vector stabilityReplace independent point-column optimization with whole-row trace-vector structure near Hilton–Milner equality.
  • Useful but insufficientIndependent trace-column optimizationUseful as a finite relaxation, but it ignores the common family rows required by a genuine trace matrix.
Selector dichotomyExact single-crown capacity, its genuine limitation, and the distinct selector-free localization route.7 displayed rows · 3 routes included
  • retained route statementExact weighted Chen-crown capacityconditional
  • retained route statementOne crown cannot charge distributed defectintermediate
  • retained route statementConcentrated defect reduces to ordinary localizationconditional
  • Research targetProve multi-crown incompatibilityopen
  • Narrowed routeMulti-crown incompatibilityThe selector branch survives only through several crowns or another configuration whose exact defect-slot colors cannot be globally reused.
  • Narrowed routeSelector-free palette coveringUse the fibre-palette covering and leakage laws; concentrated defect is routed to localization, while distributed defect still needs diversity and leakage coupling.
  • Useful but insufficientSingle Chen crownThe exact perfect-graph calculation exposes genuine tail capacity, so one crown cannot close distributed defect.
Z=2 and dense LC₂ frontierLC₁, exact shadow-and-slack systems, the five-color dense reduction, and the two immediate closing obligations.9 displayed rows · 4 routes included
  • retained route statementOne-extra-color localization
  • retained route statementExact Z=2 shadow-and-slack systemsconditional
  • retained route statementDense LC₂ five-color normal formconditional
  • Research targetProve the dense partition-aware shadow theoremopen
  • Research targetRelate blocker complexity to shadow costopen
  • Active routeDense partition-aware shadowExploit the five-way partition, completeness, monochromatic face, middle band, and codimension-two link budget simultaneously.
  • Active routeBlocker complexity versus shadow costCharge the four required units of blocker chromatic excess against the exact Z=2 shadow-and-saturation-slack budget.
  • Not yet justifiedExact triple transversalityThe earlier σ=0 conclusion used an unjustified promotion from residual-list maximality to global maximality.
  • Eliminated routeNaive LC₁ iterationRecolored vertices can split among several retained colors, so one removed color may create several localized colors.
Separate finite candidateThe challenged (11,4,45) claim, its failed partial verifier, and the explicit recovery and independent-check obligation.5 displayed rows · 1 route included
  • retained route statementCandidate χ₄₅(KG(11,4))=126special case
  • ChallengeThe integrated verifier stopped in the e=2 matching checker because a generic helper imposed an endpointwise nonpositivity assertion stronger than the required negative edge-sum condition. Equality-family definitions, tuple conventions, and explicit certificate lists are also absent from the mounted record.unsupported step · open
  • ComputationPartial integrated C++ verification attempt for the candidate χ₄₅(KG(11,4))=126.The checker compiled but stopped in the e=2 matching branch at an overstrong endpoint assertion; missing equality-family definitions and certificate lists prevent reconstruction from the current work alone. · reported unreproduced
  • Research targetComplete the independent (11,4,45) verifierblocked
  • Route held in reserveIndependent (11,4,45) verifierRecover the missing definitions, correct the checker, emit certificates, and independently reimplement the finite candidate before treating it as established.
Retained route limitsFailures that prevent repeating scalar, coordinatewise, maximality, single-crown, unrestricted peeling, ordinary topology, raw MILP, or naive recoloring arguments.16 displayed rows · 8 routes included
  • Useful failureScalar Hilton–Milner deficit countingreported failure
  • Useful failureIndependent Kruskal–Katona optimization of each trace columnreported failure
  • Useful failurePromoting residual-list maximality to global maximalityreported failure
  • Useful failureA single weighted Chen crownreported failure
  • Useful failureUnrestricted exact k-layer peelingreported failure
  • Useful failureOrdinary Lovász, colorability-defect, neighborhood-complex, Schrijver, or Fan/Tucker boundsreported failure
  • Useful failureDirect MILP over raw residual color-class variablesreported failure
  • Useful failureNaively iterate the LC₁ recoloring proofreported failure
  • Useful but insufficientScalar Hilton–Milner aloneUseful for broad elimination and k≤3, but too weak as an independent-family bound in the first unresolved k≥4 cases.
  • Useful but insufficientIndependent trace-column optimizationUseful as a finite relaxation, but it ignores the common family rows required by a genuine trace matrix.
  • Not yet justifiedExact triple transversalityThe earlier σ=0 conclusion used an unjustified promotion from residual-list maximality to global maximality.
  • Useful but insufficientSingle Chen crownThe exact perfect-graph calculation exposes genuine tail capacity, so one crown cannot close distributed defect.
  • Refuted routeUnrestricted exact-layer peelingRefuted by a nine-color 3-fold KG(5,2) example; threshold-specific or local-layer variants remain possible.
  • Useful but insufficientOrdinary topological lower boundsBase Kneser-color forcing does not yield one distinct color per defect slot at one common core.
  • Route held in reserveRaw residual-design MILPThe unsymmetrized formulation stalled under color symmetry; orbit-, trace-, blocker-, or palette-level symmetry breaking is the retained computational direction.
  • Eliminated routeNaive LC₁ iterationRecolored vertices can split among several retained colors, so one removed color may create several localized colors.
How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeRule out every complete five-class partition of the k-subsets of [2k+1] satisfying the monochromatic-face, middle-band, link-shadow, completeness, and transversal constraints.

2 approaches have already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

  • Prove that no partition satisfying all seven conditions in Section 11.12 exists.

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Prepared starting pointProve the dense partition-aware shadow theorem

Stahl’s Multichromatic Kneser-Graph Conjecture · ready to start

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Research contextPrepared context for any AI agent

When every k-subset of [n] receives s colors and disjoint subsets receive disjoint palettes, Stahl’s conjecture gives the exact minimum total number of colors.

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Sources and references6 cited works · next context review by Nov 2, 2026

The mathematical context was checked on Aug 2, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
  2. 2
    n-Tuple colorings and associated graphsoriginal source · accessed Aug 2, 2026
  3. 3
    Proof of Stahl's conjecture in some new casespeer reviewed result · accessed Aug 2, 2026
  4. 4
    A special case of the Stahl conjecturepeer reviewed result · accessed Aug 2, 2026
  5. 5
  6. 6
    n-Tuple colorings and associated graphsoriginal source · accessed Aug 2, 2026

Important qualifications

  • Do not confuse this conjecture with Herbert Stahl's Bessis-Moussa-Villani theorem or Saul Stahl's unrelated conjectures on graph embeddings.
  • Empty formalization or computation lists mean that none was verified in this scoped search, not that none exists.

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