Rota's basis conjecture

The triangles are all required to be non-degenerate, meaning that they do not have all three vertices on a line.

To see this as an instance of the basis conjecture, one may use either linear independence of the vectors (

For this linear algebra and this matroid, the bases are exactly the non-degenerate triangles.

Analogously, for points in three-dimensional Euclidean space, the conjecture states that the sixteen vertices of four non-degenerate tetrahedra of four different colors may be regrouped into four rainbow tetrahedra.

[3] For arbitrary matroids, it is possible to arrange the basis elements into a matrix the first Ω(√n) columns of which are bases.

Substantial progress on the conjecture of Bárány and Larman was made by Blagojević, Matschke & Ziegler (2009).