Math Conjectures

Rota's Basis Conjecture

OPENMajorConjectureProposed c. 1989 · Standard version

Canonical statement

Let VV be an nn-dimensional vector space over a field, and let B1,…,BnB_1,\ldots,B_n be nn (not necessarily distinct) bases of VV. It is possible to order each Bi=(bi1,…,bin)B_i=(b_{i1},\ldots,b_{in}) so that {b1j,…,bnj}\{b_{1j},\ldots,b_{nj}\} is a basis of VV for every j∈{1,…,n}j\in\{1,\ldots,n\}.
View source LaTeX
Let \(V\) be an \(n\)-dimensional vector space over a
field, and let \(B_1,\ldots,B_n\) be \(n\) (not necessarily distinct)
bases of \(V\). It is possible to order each
\(B_i=(b_{i1},\ldots,b_{in})\) so that
\(\{b_{1j},\ldots,b_{nj}\}\) is a basis of \(V\) for every
\(j\in\{1,\ldots,n\}\).
COMB-007:[RotaBasisHuangRota1994][Sauermann2024Rota]
CombinatoricsGraph theory

Notes

Rota's basis conjecture asserts that given nn bases B1,…,BnB_1,\ldots,B_n of an nn-dimensional vector space over any field, one can order each basis so that, writing them as the rows of an n×nn\times n array, every column is again a basis. The conjecture is attributed to Gian-Carlo Rota and took its modern form around 1989; it appears in print in the work of Huang and Rota, who related it to conjectures on Latin squares and to straightening coefficients [RotaBasisHuangRota1994].

Through these connections the problem sits close to sign-counting questions about Latin squares, and partial results are known over particular fields, in particular dimensions, and in highly structured regimes. In a different direction, Sauermann showed that the conjecture holds for random bases of vector spaces [Sauermann2024Rota], so typical instances are no obstruction.

Despite this progress, no argument covers arbitrary input bases over an arbitrary field, and the conjecture remains open in general; a resolution must handle worst-case collections of bases uniformly in the dimension and the field.

References (2)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.