[#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 \(B_i\).
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.
Definition 2 (Repeated vectors are treated with their multiplicities in the input collection). Repeated vectors are treated with their multiplicities in the input collection.
Remark 1. The conjecture asks whether several bases can always be reorganized into a second, transverse family of bases.
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
Current 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]
1Records
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
- Ternary representatives of finite-order integral matriceslinear algebra
- Orders of ternary row-orthogonal matrices with a full rowlinear algebra
- Hadamard matrix conjecturelinear algebra
How to cite
TheoremDB contributors, “Rota's basis conjecture,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/rotas-basis-conjectureThis page as plain text: rotas-basis-conjecture.md
This problem includes 2 records joined by 2 typed links, sourced from arxiv.org[1], current as of July 31, 2026.
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.