[#P45] Rota's basis conjecture

Problem. Let \(V\) be an \(n\)-dimensional vector space and let \(B_1,\ldots,B_n\) be pairwise disjoint bases of \(V\). Their union can be partitioned into \(n\) bases, each containing exactly one vector from every \(B_i\).

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Definitions and notation

1Context

The conjecture asks whether several bases can always be reorganized into a second, transverse family of bases.

2Problem setup

Definition 1 (Each input basis supplies one vector to each column, so every column). Each input basis supplies one vector to each column, so every column is transversal across the original bases.

Remark 1. Repeated vectors are treated with their multiplicities in the input collection.

3What counts as a solution

Prove the transversal decomposition for every n, field, and collection of n bases, or give a concrete collection and prove that no valid rearrangement exists.

1Status

Saved packet · July 31, 2026

What counts as a solution

Saved packet status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: Montgomery and Sauermann prove `(1-o(1))n` disjoint transversal bases and a cover by `(1+o(1))n` transversal bases. Sauermann proves the exact result with probability `1-o(1)` in specified random models. Exact unresolved remainder: Prove n disjoint transversal bases for every rank-n instance, or certify a counterexample.^[1][2]

1Packet records

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Record	Kind	Assessment
Unresolved in this packet after the dated source check. Strongest checked result: Montgomery and Sauermann prove `(1-o(1))n` disjoint transversal bases and a cover by `(1+o(1))n` transversal bases. Sauermann proves the exact result with probability `1-o(1)` in specified random models. Exact unresolved remainder: Prove n disjoint transversal bases for every rank-n instance, or certify a counterexample.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 Unresolved in this packet after the dated source check. Strongest checked result: Montgomery and Sauermann prove `(1-o(1))n` disjoint transversal bases and a cover by `(1+o(1))n` transversal bases. Sauermann proves the exact result with probability `1-o(1)` in specified random models. Exact unresolved remainder: Prove n disjoint transversal bases for every rank-n instance, or certify a counterexample.^[1][2] Relevance to this problem For Rota's basis conjecture, this successor gives readable dated status prose and the exact remaining research boundary. 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 packet's cited sources and equivalent formulations were checked in the dated review recorded below. Strongest checked result: Montgomery and Sauermann prove `(1-o(1))n` disjoint transversal bases and a cover by `(1+o(1))n` transversal bases. Sauermann proves the exact result with probability `1-o(1)` in specified random models. Exact unresolved remainder: Prove n disjoint transversal bases for every rank-n instance, or certify a counterexample. Open full record →
Prove the transversal decomposition for every n, field, and collection of n bases, or give a concrete collection and prove that no valid rearrangement exists.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Prove the transversal decomposition for every n, field, and collection of n bases, or give a concrete collection and prove that no valid rearrangement exists.^[1] Relevance to this problem For Rota's basis conjecture, record rotas-basis-conjecture-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Prove the transversal decomposition for every n, field, and collection of n bases, or give a concrete collection and prove that no valid rearrangement exists. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Prove the transversal decomposition for every n, field, and collection of n bases, or give a concrete collection and prove that no valid rearrangement exists. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Open full record →
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Notes and companion material

Original intake status. The cited paper calls Rota's basis conjecture wide open. The source and public status were checked on 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.

Asymptotic, random, and special-field results do not settle every collection of bases.

Computational notes

Finite-field searches can verify bounded dimensions and field sizes only.

2See also

Nonvanishing of Rudin-Shapiro Hankel determinantslinear algebra
Nonvanishing of Baum-Sweet Hankel determinantslinear algebra
Ternary representatives of finite-order integral matriceslinear algebra

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Plain text

“Rota's basis conjecture.” TheoremDB. P45. Problem statement; statement text SHA-256 34d1db69e15aa9722396391cd719040acb49b78d315c7d0ae86d8744ecb7907b. https://theoremdb.org/statement/?ref=P45

BibTeX

@misc{theoremdb-problem-34d1db69e15aa9722396391cd719040acb49b78d315c7d0ae86d8744ecb7907b,
  title = {{Rota's basis conjecture}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 34d1db69e15aa9722396391cd719040acb49b78d315c7d0ae86d8744ecb7907b},
  url = {https://theoremdb.org/statement/?ref=P45}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 2 typed links, sourced from arxiv.org^[1], current as of July 31, 2026.

1Lean verification

1References

Packet source. Lisa Sauermann, “Rota's basis conjecture holds for random bases of vector spaces”. arXiv:2203.17121 (2022). Lisa Sauermann, arXiv:2203.17121, abstract and introduction. ↗preprint · primary source · arXiv:2203.17121, checked 2026-07-31 · checked 2026-07-31Source use: original summary.The cited paper calls Rota's basis conjecture wide open. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.Also cited at Abstract and introduction.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.Packet-linked open-status and random-bases theorem.Source named by the research packet.
Richard Montgomery and Lisa Sauermann, “Asymptotically-tight packing and covering with transversal bases in Rota's basis conjecture”. arXiv:2508.05601 (2025). Abstract. ↗preprint · primary source · arXiv:2508.05601v1 · checked 2026-08-01Source use: original summary.Current asymptotically tight packing and covering theorem.

An original CC0 restatement prepared by TheoremDB maintainers.

[#P45] Rota's basis conjecture

1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

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Computational notes

2See also

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1Context

2Problem setup

3What counts as a solution

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1Packet records

Recent contributions

Computational notes

2See also

1References

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Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance