Math 2.0

Combinatorics Algebraic combinatorics Conjecture

Rota's basis conjecture

Builds on Vector spaceBasisMatroidBasis of a matroid

Statement

Let VV be a vector space of dimension n≥1n \ge 1 over a field. Let B1,…,BnB_1, \dots, B_n be bases of VV, regarded as nn disjoint labelled families of nn vectors. Then the n2n^2 vectors can be arranged in an n×nn \times n grid with the following properties. For each ii, the ii-th row consists of the vectors of BiB_i. Each column is a basis of VV.

Remarks

The conjecture is due to Gian-Carlo Rota, and it is open. It has a natural form for matroids: given nn bases of a matroid of rank nn, the elements can be arranged in such a grid with each column a basis. For n=2n = 2 the statement is easy to check by hand: with B1={a,b}B_1 = \set{a, b} and B2={c,d}B_2 = \set{c, d}, at least one of the pairings {a,c},{b,d}\set{a, c}, \set{b, d} or {a,d},{b,c}\set{a, d}, \set{b, c} gives two bases. Huang and Rota related the vector space form to the Alon–Tarsi conjecture on Latin squares, which gives the conjecture for some even nn over fields of characteristic zero. Asymptotic versions are known that produce many disjoint column bases, but not nn of them in general.

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Basis, concept, no evidence levelBasisconceptBasis of a matroid, concept, no evidence levelBasis of a matroidconceptMatroid, concept, no evidence levelMatroidconceptVector space, concept, no evidence levelVector spaceconceptRota's basis conjecture, conjecture, no evidence levelRota's basis conjectu…conjecture
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  1. Basis concept
  2. Basis of a matroid concept
  3. Matroid concept
  4. Vector space concept
  5. Rota's basis conjecture conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Rota%27s_basis_conjecture
About this page
TagD8ZD10YF3Y
URIm2://D8ZD10YF3Y
Kindconjecture
WritingW0 stub
MSC05B35, 15A03
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AboutBasis, Matroid
UsesVector space, Basis, Matroid, Basis of a matroid
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Changed2 November 2026

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