Rota's Basis Conjecture for Matroids with Density Close to One

TL;DR

This paper investigates Rota's basis conjecture in matroids with nearly full density, demonstrating the construction of multiple disjoint rainbow bases under specific conditions related to the matroid's size.

Contribution

The paper extends Rota's basis conjecture to matroids with density close to one, providing bounds on the number of disjoint rainbow bases that can be constructed.

Findings

Constructs n - k^3 disjoint rainbow bases in certain dense matroids.

Shows the relationship between matroid size and the number of rainbow bases.

Provides bounds depending only on the parameter k.

Abstract

Rota's basis conjecture (RBC) states that given a collection B of n bases in a matroid M of rank n, one can always find n disjoint rainbow bases with respect to B. We show that if M is a matroid having n + k elements, then one can construct n - k^3 disjoint rainbow bases, where b is a constant depending only on k.