ProofAtlas

Matroid theory · transversal bases · combinatorial exchange

Rota’s Basis Conjecture

Collaboration beta

Given n disjoint bases of a rank-n matroid, can all n² elements always be rearranged into n new bases, each taking exactly one element from every original basis? The source advances one conditional branch but explicitly leaves the general conjecture open.

|Tj∩Bi|=1(1≤i,j≤n)
Known results and sources
Landscape card showing four disjoint matroid bases rearranged into four color-coded transversal groups, with the conjecture marked open.
Rota’s Basis Conjecture asks whether n disjoint rank-n bases can always be repartitioned into n transversal bases.

Research problem

Exact mathematical statement

Let M be a rank-n matroid and let B_1,…,B_n be n pairwise disjoint bases. Does their union admit a partition into n bases T_1,…,T_n such that each T_j contains exactly one element from every B_i?

{B1,…,Bn}⟶{T1,…,Tn},|Tj∩Bi|=1.\{B_1,…,B_n\}\longrightarrow\{T_1,…,T_n\},\qquad |T_j\cap B_i|=1.

Revision v8 explicitly reports no proof or counterexample for unrestricted Rota’s Basis Conjecture. Its q≤10 conclusions are conditional project theorems in a specific parallel-extension/simple-sparse-paving branch and remain audit-required.

Problem infographic

Problem at a glance

Three-panel landscape explainer showing four disjoint input bases, their row-rainbow rearrangement, and the open question whether every output is a basis.
A transversal takes one element from each input basis; the universal question is whether all n transversals can simultaneously be bases.

Current mathematical picture

Where work on Rota’s Basis Conjecture stands

Open conjecture

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureSource-reported limitation

The following claim is rejected or insufficient in the recorded route: Low-excess closure in the conditional branch is not a proof for arbitrary matroids, and the seven-row p=4 absorber must not be generalized to arbitrary ambient row sets. Audit every projection, at-most-one-bad, terminal-orientation, transport, cocircuit-maximality, and verifier interface; separately advance the same conditional branch to q=11, beginning with rank 7.

Route status · Narrowed route
Main reductionCurrent reduction

Within the stated branch, the source represents parallel extensions of simple sparse-paving matroids by a row–simplification-point multigraph, seeks a proper edge coloring with at most one bad projected support, and uses terminal cube orientations, mismatch budgets, transport, absorbers, and finite certificates to eliminate low excess.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeAudit the load-bearing branch interfacesTask status · Ready to work on

Work mapped so far

Rota’s Basis Conjecture in numbers

3.8kretained lines of mathematical investigation3,818 in the current working snapshot
Argument development
3,012 · 79%
Explored or eliminated routes
262 · 7%
Computational analysis
115 · 3%
Open obligations
162 · 4%
Definitions and setup
267 · 7%
7selected mapped statements1routes investigated5open questions5contribution-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.

Proposed byGian-Carlo RotaProposed1989Status checkedAug 14, 2026
Source recordedAug 14, 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

13 selected steps

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

13 selected steps

Scroll horizontally to explore the route

Working route overview for Rota’s Basis ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Can n disjoint matroid bases always be rearranged into n transversal bases? — Depends on missing premiseCan n disjoint matroid basesalways be rearranged into ntransversal…Conditional low-excess ledger — Depends on missing premiseConditional low-excessledgerCurrent reduction — Depends on missing premiseCurrent reductionClosing target — Depends on missing premiseClosing targetExact transversal-basis target — Depends on missing premiseExact transversal-basistargetReported rank-12 endpoint — Depends on missing premiseReported rank-12 endpointReported rank-8 endpoint — Depends on missing premiseReported rank-8 endpointSource-reported limitation — stoppedSource-reported limitationAudit the load-bearing branch interfaces — OpenAudit the load-bearingbranch interfacesIndependently reproduce finite certificate scopes — OpenIndependently reproducefinite certificate scopesAdvance the q=11 rank-7 case — OpenAdvance the q=11 rank-7 caseUnrestricted conjecture remains open — OpenUnrestricted conjectureremains openPublication-level audit — OpenPublication-level audit
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.

Explored alternatives

Other routes

1 recorded
Narrowed routeSource-reported limitation

The following claim is rejected or insufficient in the recorded route: Low-excess closure in the conditional branch is not a proof for arbitrary matroids, and the seven-row p=4 absorber must not be generalized to arbitrary ambient row sets. Audit every projection, at-most-one-bad, terminal-orientation, transport, cocircuit-maximality, and verifier interface; separately advance the same conditional branch to q=11, beginning with rank 7.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

5 featured tasks
01
Audit the load-bearing branch interfacesSuggested move: Check the projection, coloring, terminal-orientation, transport, and cocircuit-maximality arrows against exact hypotheses and record independent review evidence.
Ready to work on
02
Independently reproduce finite certificate scopesSuggested move: Rebuild the intended quotient universes and verify that every accepted support uses only independently certified bases, without executing the private packet in this staging lane.
Ready to work on
03
Unrestricted conjecture remains open

The current work claims neither a complete proof nor a counterexample for unrestricted Rota’s Basis Conjecture.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
Publication-level audit

All branch-wide theorem statements remain audit-required until the listed dependency and verifier questions are checked.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
05
Advance the q=11 rank-7 caseSuggested move: After audit separation, classify q=11 forced-support atoms at rank 7 and test whether the conditional transport and absorber interfaces extend without hidden assumptions.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026
Current statusOpen conjecture

The exact universal conjecture remains open. Current primary-source progress includes random-basis theorems and asymptotically tight packing and covering results, none of which supplies the exact partition into n transversal bases for every rank-n matroid.

[2][3][4]
External progress

What the literature has established

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

  1. PreprintMontgomery and Sauermann reported asymptotically tight packing of (1-o(1))n disjoint transversal bases and covering by (1+o(1))n transversal bases, without closing the exact n-by-n partition.[4]
  2. Peer reviewedPokrovskiy proved an asymptotic relaxation producing n-o(n) disjoint rainbow independent sets of size n-o(n).[3]
  3. PreprintSauermann proved that the conjecture holds with probability tending to one for specified random-basis models, while describing the exact conjecture as wide open.[2]
  4. Historical sourceThe conjecture was attributed to Rota in 1989 and published in Huang and Rota’s 1994 work connecting it with Latin-square conjectures.[1][4]
4 cited sources3 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusRota’s Basis Conjecture
Weaker or relaxed formasymptotic Rota basis packing

Asymptotic packing asks for almost n disjoint transversal bases or almost-full rainbow independent sets rather than an exact partition into n bases.

[3][4]
Solved special caserandom bases over finite or bounded-entry models

Specified random-basis models satisfy the exact conjecture with probability tending to one as n grows; this is not a worst-case theorem.

[2]
Weaker or relaxed formtransversal-basis covering

The covering variant permits a little more than n transversal bases to cover all input elements and is asymptotically tight.

[4]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

  • Formalization targetA checked statement must encode the exact rank-n matroid quantifiers, n disjoint input bases, and n output bases with one element from each input basis.
  • Formalization targetAny formal account must separate vector-space special cases, random-input theorems, asymptotic relaxations, paving or sparse-paving cases, and the unrestricted matroid conjecture.
  • Formalization targetThe private packet’s projection, coloring, transport, maximality, and finite-certificate interfaces require exact independent review before formal evidence claims.

Detailed research inventory

Claims, milestones, and routes in the current map

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

5 standing statements2 proposed statements5 open questions1 narrowed routes
Statements by mathematical role7 selected mapped statements
  • theorem candidate1 of 71
  • reduction2 of 72
  • lemma4 of 74
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.21 displayed rows · 1 route included
  • retained route statementCan n disjoint matroid bases always be rearranged into n transversal bases?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementExact transversal-basis targetintermediate
  • retained route statementConditional low-excess ledgerintermediate
  • retained route statementReported rank-12 endpointintermediate
  • retained route statementReported rank-8 endpointintermediate
  • Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
  • Useful failureSource-reported limitationreported failure
  • Research targetAudit the load-bearing branch interfacesopen
  • Research targetIndependently reproduce finite certificate scopesopen
  • Research targetAdvance the q=11 rank-7 caseopen
  • Research targetUnrestricted conjecture remains openopen
  • Research targetPublication-level auditopen
  • ComputationThe current work reports exact finite verifiers and rerun logs for selected absorber, pair-atom, and Kempe-state searches; these attachments were not executed during this staging work.The source reports that its embedded certificates close the specified low-rank and q=10 subcases, while explicitly retaining independent-enumerator and publication-audit limitations. · reported unreproduced
  • Narrowed routeSource-reported limitationThe following claim is rejected or insufficient in the recorded route: Low-excess closure in the conditional branch is not a proof for arbitrary matroids, and the seven-row p=4 absorber must not be generalized to arbitrary ambient row sets. Audit every projection, at-most-one-bad, terminal-orientation, transport, cocircuit-maximality, and verifier interface; separately advance the same conditional branch to q=11, beginning with rank 7.
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 bridgeAudit the load-bearing branch interfaces

1 approach has 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.

  • Supply a complete argument with every imported premise identified.
  • Survive an independent attempt to falsify the proposed step.

Continue the mathematics

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Prepared starting pointAudit the load-bearing branch interfaces

Rota’s Basis Conjecture · ready to start

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Research contextPrepared context for any AI agent

Given n disjoint bases of a rank-n matroid, can all n² elements always be rearranged into n new bases, each taking exactly one element from every original basis? The source advances one conditional branch but explicitly leaves the general conjecture open.

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references4 cited works · next context review by Nov 14, 2026

The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    On the relations of various conjectures on Latin squares and straightening coefficientsoriginal source · Rosa Huang, Gian-Carlo Rota · Discrete Mathematics 128, 225–236 · 1994-04-15 · DOI 10.1016/0012-365X(94)90114-7 · accessed Aug 14, 2026
  2. 2
    Rota’s basis conjecture holds for random bases of vector spacespreprint · Lisa Sauermann · arXiv · 2022-03-31 · ARXIV 2203.17121 · accessed Aug 14, 2026
  3. 3
    Rota’s Basis Conjecture holds asymptoticallypeer reviewed result · Alexey Pokrovskiy · Proceedings of the American Mathematical Society 153, 4101–4118 · 2025 · ARXIV 2008.06045 · DOI 10.1090/proc/16581 · accessed Aug 14, 2026
  4. 4
    Asymptotically-tight packing and covering with transversal bases in Rota’s basis conjecturepreprint · Richard Montgomery, Lisa Sauermann · arXiv · 2025-08-07 · ARXIV 2508.05601 · accessed Aug 14, 2026

Important qualifications

  • This source-separated metadata does not verify any mathematical claim in the private packet or grant review, credit, publication, or deployment authority.
  • The cited asymptotic and random-input theorems do not prove the exact universal partition conjecture.
  • The current work’s parallel-extension/simple-sparse-paving reduction and q≤10 ledger require separate statement-level and dependency-level review.
  • No checked formalization of the exact universal matroid statement was identified in this bounded collection; absence from this collection is not proof that none exists.

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