Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Bounds on parameters of minimally non-linear patterns   Let $ex(n, P)$ be the maximum possible number of ones in any 0-1 matrix of
dimensions $n \times n$ that avoids $P$. Matrix $P$ is called minimally
non-linear if $ex(n, P) = \omega(n)$ but $ex(n, P') = O(n)$ for every strict
subpattern $P'$ of $P$. We prove that the ratio between the length and width of
any minimally non-linear 0-1 matrix is at most $4$, and that a minimally
non-linear 0-1 matrix with $k$ rows has at most $5k-3$ ones. We also obtain an
upper bound on the number of minimally non-linear 0-1 matrices with $k$ rows.
  In addition, we prove corresponding bounds for minimally non-linear ordered
graphs. The minimal non-linearity that we investigate for ordered graphs is for
the extremal function $ex_{<}(n, G)$, which is the maximum possible number of
edges in any ordered graph on $n$ vertices with no ordered subgraph isomorphic
to $G$.

30
The categories of this paper are: q-alg
4
The categories of this paper are: cond-mat
25
The categories of this paper are: patt-sol
33
The categories of this paper are: plasm-ph
0
The categories of this paper are: hep-ph
36
The categories of this paper are: cmp-lg
9
The categories of this paper are: nlin
20
The categories of this paper are: acc-phys
15
The categories of this paper are: nucl-ex
22
The categories of this paper are: chao-dyn
18
The categories of this paper are: econ
21
The categories of this paper are: adap-org
32
The categories of this paper are: atom-ph
5
The categories of this paper are: gr-qc
16
The categories of this paper are: stat
3
The categories of this paper are: physics