  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.
