Improved bounds for Rota's Basis Conjecture

We prove that, if $B_1,\dots ,B_n$ are disjoint bases of a rank-$n$ matroid, then there are at least $\lfloor \frac{n}{6\lceil \log n\rceil }\rfloor$ disjoint transversals of $(B_1,\dots ,B_n)$ that are also bases.