Cyclic orderings of matroids

Definitions and mathematical statement

A finite matroid $M=(S,\mathcal I)$ consists of a family of independent subsets containing $\varnothing$, closed under subsets, and satisfying exchange: if $I,J\in\mathcal I$ and $|I|<|J|$, some $e\in J\setminus I$ has $I\cup\{e\}\in\mathcal I$. Its bases are the maximal independent sets; all have the same cardinality $r$.

Suppose $r\ge1$ and the ground set is a disjoint union of $k$ bases, so $|S|=kr$. Must there be an ordering $s_1,\ldots,s_{kr}$ of all elements such that \[ \{s_j,s_{j+1},\ldots,s_{j+r-1}\} \text{ is a basis for every }j, \] where subscripts are read modulo $kr$? The required order is cyclic, including windows that cross its chosen starting point.

The given basis partition only guarantees independence for $k$ disjoint blocks. The conclusion requires independence for all overlapping windows of length $r$. The cyclic order may completely rearrange the original bases; it need not preserve their internal order or keep them as consecutive blocks. The question ranges over arbitrary finite matroids, without assuming a representation over a field. A theorem for graphic or another representable subclass therefore establishes only a special case of this universal assertion.

Short English statement

If a matroid's ground set splits into bases, can all its elements be cyclically arranged so that every rank-sized consecutive block is a basis?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3000024 (AMR-029-0024), 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: Cyclic orderings of matroids. Original source indexed by AMR-029-0024. https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems.
• S3 EGRES Open, Cyclic orderings of matroids. Individual source page and explanatory remarks. https://lemon.cs.elte.hu/egres/open/Cyclic_orderings_of_matroids.