Richard Montgomery and Lisa Sauermann Present Major Progress on Rota’s Basis Conjecture

Combinatorics and more 2025-09-09

Richard Montgomery and Lisa Sauermann: Asymptotically-tight packing and covering with transversal bases in Rota’s basis conjecture

Abstract: 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.

Short commentary: This is a major breakthrough on the beautiful Rota’s basis conjecture. A classic paper on the subject is the 1989 paper by Rosa Huang and Gian Carlo-Rota. See this post for several famous conjectures by Rota, and this post about the related Alon-Tarsi conjecture. Rota’s basis conjecture extends Dinitz’ conjecture about Latin squares that was solved by Galvin, see this post.

In 2017 Timothy Chow lunched a polymath project (Polymath 12) to solve it. (Here is my post on the project with various variants of the conjecture, the first post on the polymath blog, and the wiki). Timothy Chow also asked the third most popular “Test Your Intuition” question over my blog. (Come to think of it, I don’t remember if and when Timothy and I met in person.)

In 2020 Alex Pokrovskiy proved that it is possible to find n − o(n) disjoint rainbow independent sets of size n − o(n). Bucić, Kwan, Pokrovskiy, and Sudakov proved that it is possible to find $n/2-o(n)$ disjoint rainbow basis and the new result by Richard and Lisa shows that it is possible to find $n-o(n)$ disjoint rainbow basis!

h/t Benny Sudalov