Omni-MATH / Let n(≥2) be a positive integer. Find the minimum m, so that there exists x_(ij)(1≤i,j≤n) satisfying: (1)For every 1≤i,j≤n,x_(ij)=max{x_(i1),x_(i2),...,x_(ij)} or …

Let n(≥2) be a positive integer. Find the minimum m, so that there exists x_(ij)(1≤i,j≤n) satisfying:
(1)For every 1≤i,j≤n,x_(ij)=max{x_(i1),x_(i2),...,x_(ij)} or x_(ij)=max{x_(1j),x_(2j),...,x_(ij)}.
(2)For every 1≤i≤n, there are at most m indices k with x_(ik)=max{x_(i1),x_(i2),...,x_(ik)}.
(3)For every 1≤j≤n, there are at most m indices k with x_(kj)=max{x_(1j),x_(2j),...,x_(kj)}.

Let $ n(\ge2) $ be a positive integer. Find the minimum $ m $, so that there exists $x_{ij}(1\le i ,j\le n)$ satisfying:
(1)For every $1\le i ,j\le n, x_{ij}=max\{x_{i1},x_{i2},...,x_{ij}\} $ or $ x_{ij}=max\{x_{1j},x_{2j},...,x_{ij}\}.$
(2)For every $1\le i \le n$, there are at most $m$ indices $k$ with $x_{ik}=max\{x_{i1},x_{i2},...,x_{ik}\}.$
(3)For every $1\le j \le n$, there are at most $m$ indices $k$ with $x_{kj}=max\{x_{1j},x_{2j},...,x_{kj}\}.$