# Rota's conjecture on disjoint bases

Let [math]M[/math] be a matroid of rank *n* whose ground set *S* can be partitioned into *n* disjoint bases [math]B_1,\dots,B_n[/math]. Is it true that [math]B_1,\dots,B_n[/math] always have *n* disjoint transversals that are bases of [math]M[/math]?

## Remarks

This conjecture was made by Rota in 1989, see ^{[1]}. Geelen and Humphries ^{[2]} proved it for paving matroids.

For general matroids, Geelen and Webb ^{[3]} proved that [math]B_1,\dots,B_n[/math] always have [math]\Omega(\sqrt{n})[/math] disjoint transversals that are bases. This was recently improved to [math](1/2 − o(1))n[/math] by Bucić, Kwan, Pokrovskiy, and Sudakov ^{[4]}. On the other hand, it follows from the results of Aharoni and Berger ^{[5]} that *S* can be partitioned into *2n* common independent sets.

## Related questions

## See also

Rota's conjecture at the Open Problem Garden, Polymath page

## References

- ↑ R. Huang, G.-C. Rota,
*On the relations of various conjectures on Latin squares and straightening*coefficients*(1994), DOI link* - ↑ J. Geelen, P.J. Humphries,
*Rota's basis conjecture for paving matroids*, SIAM J. Discrete Math. 20 (2006), 1042-1045. DOI link, Citeseer link - ↑ J. Geelen, K. Webb,
*On Rota's basis conjecture*, SIAM J. Discrete Math. 21 (2007), 802-804. DOI link - ↑ M. Bucić, M. Kwan, A. Pokrovskiy, B. Sudakov,
*Halfway to Rota’s basis conjecture*(2018), arXiv link - ↑ R. Aharoni, E. Berger,
*The intersection of a matroid and a simplicial complex*, Trans. Amer. Math. Soc. 358 (2006), 4895-4917. Journal link