Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Girth conditions and Rota's basis conjecture   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.

30
The categories of this paper are: q-alg
3
The categories of this paper are: physics
36
The categories of this paper are: cmp-lg
13
The categories of this paper are: quant-ph
18
The categories of this paper are: econ
24
The categories of this paper are: dg-ga
5
The categories of this paper are: gr-qc
7
The categories of this paper are: hep-th
6
The categories of this paper are: astro-ph
33
The categories of this paper are: plasm-ph
8
The categories of this paper are: hep-ex
17
The categories of this paper are: q-fin
19
The categories of this paper are: eess
1
The categories of this paper are: cs
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The categories of this paper are: alg-geom
25
The categories of this paper are: patt-sol