Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sharper bounds and structural results for minimally nonlinear 0-1
  matrices   The extremal function $ex(n, P)$ is the maximum possible number of ones in
any 0-1 matrix with $n$ rows and $n$ columns that avoids $P$. A 0-1 matrix $P$
is called minimally non-linear if $ex(n, P) = \omega(n)$ but $ex(n, P') = O(n)$
for every $P'$ that is contained in $P$ but not equal to $P$.
  Bounds on the maximum number of ones and the maximum number of columns in a
minimally non-linear 0-1 matrix with $k$ rows were found in (CrowdMath, 2018).
In this paper, we improve the bound on the maximum number of ones in a
minimally non-linear 0-1 matrix with $k$ rows from $5k-3$ to $4k-4$. As a
corollary, this improves the upper bound on the number of columns in a
minimally non-linear 0-1 matrix with $k$ rows from $4k-2$ to $4k-4$.
  We also prove that there are not more than four ones in the top and bottom
rows of a minimally non-linear matrix and that there are not more than six ones
in any other row of a minimally non-linear matrix. Furthermore, we prove that
if a minimally non-linear 0-1 matrix has ones in the same row with exactly $d$
columns between them, then within these columns there are at most $2d-1$ rows
above and $2d-1$ rows below with ones.

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The categories of this paper are: comp-gas
3
The categories of this paper are: physics
32
The categories of this paper are: atom-ph
12
The categories of this paper are: nucl-th
14
The categories of this paper are: math-ph
24
The categories of this paper are: dg-ga
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The categories of this paper are: supr-con
30
The categories of this paper are: q-alg
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The categories of this paper are: solv-int
11
The categories of this paper are: hep-lat
33
The categories of this paper are: plasm-ph
36
The categories of this paper are: cmp-lg
5
The categories of this paper are: gr-qc
13
The categories of this paper are: quant-ph
1
The categories of this paper are: cs
29
The categories of this paper are: funct-an