# Scrambled Rota conjecture

From Egres Open

Let [math]M=(S,r)[/math] be a loopless matroid of rank *k* whose ground set can be partitioned into *k* bases. Is it true that no matter how we partition *S* into sets of size *k*, the partition will have *k-1* disjoint transversals that are bases?

## Remarks

This was proposed by Aharoni and Kotlar ^{[1]}. They give a simple example showing that *k-1* cannot be replaced by *k* in the conjecture. A somewhat similar problem is the special case of Upper bound on common independent set cover when one of the matroids is a partition matroid. See also the survey of Aharoni, Charbit and Howard ^{[2]} that contains several related questions.

## References

- ↑ R. Aharoni, D. Kotlar,
*A weak version of Rota's basis conjecture for odd dimensions*, DOI link, arXiv link - ↑ R. Aharoni, P. Charbit and D. Howard,
*On a Generalization of the Ryser-Brualdi-Stein Conjecture*, DOI link, arXiv link