Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Level Matrices   Let $n>1$ and $k>0$ be fixed integers. A matrix is said to be level if all
its column sums are equal. A level matrix with $m$ rows is called reducible if
we can delete $j$ rows, $0<j<m$, so that the remaining matrix is level. We ask
if there is a minimum integer $\ell=\ell(n,k)$ such that for all $m>\ell$, any
$m\times n$ level matrix with entries in $\{0,\ldots,k\}$ is reducible. It is
known that $\ell(2,k)=2k-1$. In this paper, we establish the existence of
$\ell(n,k)$ for $n\geq 3$ by giving upper and lower bounds for it. We then
apply this result to bound the number of certain types of vector space
multipartitions.

25
The categories of this paper are: patt-sol
35
The categories of this paper are: mtrl-th
17
The categories of this paper are: q-fin
1
The categories of this paper are: cs
36
The categories of this paper are: cmp-lg
5
The categories of this paper are: gr-qc
28
The categories of this paper are: alg-geom
23
The categories of this paper are: solv-int
33
The categories of this paper are: plasm-ph
29
The categories of this paper are: funct-an
30
The categories of this paper are: q-alg
12
The categories of this paper are: nucl-th
24
The categories of this paper are: dg-ga
37
The categories of this paper are: supr-con
19
The categories of this paper are: eess
7
The categories of this paper are: hep-th