# Girth conditions and Rota's basis conjecture

## TL;DR

This paper extends results on Rota's basis conjecture by showing near-complete disjoint rainbow bases can be found in matroids with high girth and limited element overlap, using cascade techniques.

## Contribution

It establishes a near-complete version of Rota's basis conjecture for matroids with high girth and bounded element frequency, generalizing previous special cases.

## Key findings

- Achieves $n - o(n)$ disjoint rainbow bases under given conditions
- Extends prior work on paving matroids and disjoint collections
- Utilizes cascade method for matroid basis partitioning

## Abstract

Rota's basis conjecture (RBC) states that given a collection $\mathcal{B}$ of $n$ bases in a matroid $M$ of rank $n$, one can always find $n$ disjoint rainbow bases with respect to $\mathcal{B}$. In this paper, we show that if $M$ has girth at least $n-o(\sqrt{n})$, and no element of $M$ belongs to more than $o(\sqrt{n})$ bases in $\mathcal{B}$, then one can find at least $n - o(n)$ disjoint rainbow bases with respect to $\mathcal{B}$. This result can be seen as an extension of the work of Geelen and Humphries, who proved RBC in the case where $M$ is paving, and $\mathcal{B}$ is a pairwise disjoint collection. We make extensive use of the cascade idea introduced by Buci\'c et al.