Improved bounds for Rota's Basis Conjecture

TL;DR

This paper establishes a new lower bound on the number of disjoint basis transversals in a matroid, improving understanding of Rota's Basis Conjecture by providing quantitative bounds.

Contribution

It introduces a novel lower bound on the number of disjoint basis transversals in a matroid, advancing the theoretical understanding of Rota's Basis Conjecture.

Findings

At least loor(n/(6 eil(log n)))isjoint basis transversals exist

Provides a logarithmic factor in the lower bound

Advances the theoretical bounds related to Rota's Basis Conjecture

Abstract

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.