Asymptotically-tight packing and covering with transversal bases in Rota's basis conjecture

In 1989, Rota conjectured that, given any $n$ bases $B_1,\dots,B_n$ of a vector space of dimension $n$, or more generally a matroid of rank $n$, it is possible to rearrange these into $n$ disjoint transversal bases. Here, a transversal basis is a basis consisting of exactly one element from each of the original bases $B_1,\dots,B_n$. Two natural approaches to this conjecture are, to ask in this setting a) how many disjoint transversal bases can we find and b) how few transversal bases do we need to cover all the elements of $B_1,\dots,B_n$? In this paper, we give asymptotically-tight answers to both of these questions. For a), we show that there are always $(1-o(1))n$ disjoint transversal bases, improving a result of Bucić, Kwan, Pokrovskiy, and Sudakov that $(1/2-o(1))n$ disjoint transversal bases always exist. For b), we show that $B_1\cup\dots \cup B_n$ can be covered by $(1+o(1))n$ transversal bases, improving a result of Aharoni and Berger using instead $2n$ transversal bases, and a subsequent result of the Polymath project on Rota's basis conjecture using $2n-2$ transversal bases.