Rota's conjecture on disjoint bases

Definitions and mathematical statement

Let $M=(S,\mathcal I)$ be a finite matroid of rank $n\ge1$. Its independent sets are closed under taking subsets and satisfy exchange: an independent set of smaller size can be enlarged using an element of a larger independent set. Bases are independent sets of size $n$.

Suppose $S$ is partitioned into $n$ bases $B_1,\ldots,B_n$. Rota's basis conjecture asks whether $S$ can also be partitioned into bases $C_1,\ldots,C_n$ with \[ |B_i\cap C_j|=1 \quad(1\le i,j\le n). \] Equivalently, write the given bases as the rows of an $n\times n$ array and permute entries within each row so that every column is also a basis.

The new bases must be pairwise disjoint, so each of the $n^2$ ground elements is used exactly once. Finding one transversal basis is weaker than constructing the complete partition. The assertion is for arbitrary matroids; a representation over a particular field is not assumed. In a represented example, ground elements remain separately labelled even when their representing vectors agree or are parallel. The row partition is part of the hypothesis and may not be replaced by arbitrary $n$-element sets, which need not themselves be bases.

Short English statement

Can an array whose rows are disjoint matroid bases always be reordered within rows so that every column is also a basis?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3000072 (AMR-029-0072), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Egres Open - List of open problems, Open-problem page: Rota's conjecture on disjoint bases. Original source indexed by AMR-029-0072. https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems.
• S3 EGRES Open, Rota's conjecture on disjoint bases. Individual source page and explanatory remarks. https://lemon.cs.elte.hu/egres/open/Rota%27s_conjecture_on_disjoint_bases.