Halfway to Rota's basis conjecture

TL;DR

This paper advances the understanding of Rota's basis conjecture by proving that at least nearly half of the bases can be constructed as disjoint transversal bases, improving previous bounds significantly.

Contribution

The paper proves that at least approximately half of the bases can be formed as disjoint transversal bases, improving the known lower bounds for Rota's basis conjecture.

Findings

Established a bound of (1/2 - o(1))n disjoint transversal bases

Extended results to the more general setting of matroids

Improved upon the previous bound of Ω(n / log n)

Abstract

In 1989, Rota made the following conjecture. Given $n$ bases $B_{1},\dots,B_{n}$ in an $n$-dimensional vector space $V$, one can always find $n$ disjoint bases of $V$, each containing exactly one element from each $B_{i}$ (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 $\left(1/2-o\left(1\right)\right)n$ disjoint transversal bases, improving on the previous best bound of $\Omega\left(n/\log n\right)$. Our results also apply to the more general setting of matroids.