In 1989, Rota made the following conjecture. Given n bases B1, . . . , Bn in an n-dimensional
vector space V , one can always find n disjoint bases of V , each containing exactly one element
from each Bi (we call such bases transversal bases). Rota’s basis conjecture remains wide open
despite its apparent simplicity and the efforts of many researchers (for example, the conjecture
was recently the subject of the collaborative “Polymath” project). In this paper we prove that
one can always find (1/2 − o(1))n disjoint transversal bases, improving on the previous best
bound of Ω(n/ log n). Our results also apply to the more general setting of matroids
