On Rota's Basis Conjecture

Abstract

Rota conjectured that if $(B_1,\ldots,B_n)$ are disjoint bases in a rank-n matroid M, then there are n disjoint transversals of $(B_1,\ldots,B_n)$ that are bases of M. We prove the weaker result that there are $O(\sqrt n)$ disjoint transversals of $(B_1,\ldots,B_n)$ that are bases. We also prove that if $(B_1,\ldots,B_k)$ are disjoint bases of a rank-n matroid with $n> \binom{k+1}{2}$, then there are n disjoint independent transversals of $(B_1,\ldots,B_k)$.