Scrambled Rota conjecture

Definitions and mathematical statement

Let $M=(S,\mathcal I)$ be a finite loopless matroid of rank $k\ge2$. Independent sets form a hereditary family satisfying exchange; bases are independent sets of size $k$. Looplessness means every singleton is independent. Suppose the ground set can be partitioned into $k$ bases, and hence has $k^2$ elements.

Now choose an arbitrary partition $S=A_1\sqcup\cdots\sqcup A_k$ with $|A_i|=k$. These new classes need not be independent. The scrambled Rota conjecture asks for $k-1$ pairwise disjoint bases $T_1,\ldots,T_{k-1}$ satisfying \[ |T_j\cap A_i|=1 \quad(1\le i\le k,\ 1\le j\le k-1). \] Each requested basis is thus a full transversal of the new partition.

The original basis partition only certifies that the matroid has enough disjoint bases; it need not have any alignment with the new classes $A_i$. The $k-1$ transversals together use all but one element of each class. The remaining $k$ elements are not required to be a basis. Requesting $k$ disjoint transversal bases for arbitrary new classes would be a stronger assertion than this conjecture. No assumption of representability or additional independence of the scrambled classes is included. The omitted case $k=1$ is vacuous because it asks for zero transversals.

Short English statement

After arbitrarily regrouping the elements of a matroid that splits into $k$ bases, must there still be $k-1$ disjoint transversal bases?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3000076 (AMR-029-0076), 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: Scrambled Rota conjecture. Original source indexed by AMR-029-0076. https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems.
• S3 EGRES Open, Scrambled Rota conjecture. Individual source page and explanatory remarks. https://lemon.cs.elte.hu/egres/open/Scrambled_Rota_conjecture.