# matroid

## Rota's basis conjecture ★★★

Author(s): Rota

Conjecture   Let be a vector space of dimension and let be bases. Then there exist disjoint transversals of each of which is a base.

Keywords: base; latin square; linear algebra; matroid; transversal

## Aharoni-Berger conjecture ★★★

Author(s): Aharoni; Berger

Conjecture   If are matroids on and for every partition of , then there exists with which is independent in every .

Keywords: independent set; matroid; partition

## Bases of many weights ★★★

Author(s): Schrijver; Seymour

Let be an (additive) abelian group, and for every let .

Conjecture   Let be a matroid on , let be a map, put and . Then

Keywords: matroid; sumset; zero sum

## Rota's unimodal conjecture ★★★

Author(s): Rota

Let be a matroid of rank , and for let be the number of closed sets of rank .

Conjecture   is unimodal.
Conjecture   is log-concave.

Keywords: flat; log-concave; matroid