# P45: Rota's basis conjecture

- ID: `P45`
- Reference: `rotas-basis-conjecture`
- Page: https://theoremdb.org/statements/P45
- Record maturity: Reviewed problem with recorded work

## 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 \(B_i\).

### Context

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

### Problem setup

- **Definition (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 (Repeated vectors are treated with their multiplicities in the input collection).** Repeated vectors are treated with their multiplicities in the input collection.
- **Remark.** The conjecture asks whether several bases can always be reorganized into a second, transverse family of bases.

### What 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.

## Status

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](#reference-1) [2](#reference-2)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

- Original intake status: 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.
- The vector-space formulation and status were checked against the cited paper on 2026-07-22.
- Asymptotic, random, and special-field results do not settle every collection of bases.

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Computational notes

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

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `rotas-basis-conjecture`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>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 https://arxiv.org/abs/2203.17121
   - Also cited at Abstract and introduction
   - Also cited at Editorial research route recorded 2026-07-31
   - preprint; primary source; arXiv:2203.17121, checked 2026-07-31; checked 2026-07-31
   - Source 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.
   - 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.
2. <a id="reference-2"></a>Richard Montgomery and Lisa Sauermann, “Asymptotically-tight packing and covering with transversal bases in Rota's basis conjecture”. arXiv:2508.05601 (2025). Abstract https://arxiv.org/abs/2508.05601
   - preprint; primary source; arXiv:2508.05601v1; checked 2026-08-01
   - Source use: original_summary
   - Current asymptotically tight packing and covering theorem.