Largest possible 2D combination not containing 2 common rectangles This picture made me wonder what is the largest common rectangle in the picture below ( if any ( without allowing rotation)?
Next question given n distinct letters what is the largest possible 2d covering ( i.e. putting letters in a grid ) before there are 2 common rectangles of one with side having length of n letters.
Is it possible to create arbitrary large 2d grids without having 2 common squares of sides $n^2$ length?
I guess these are questions analogous to greatest common sequence.
I could only find $ \begin{array}{lcr}
\mbox O & O \\
\mbox O & G \\\end{array}
$  

 A: I think there is one piece that is easy to answer, 
"Is it possible to create arbitrary large 2d grids without having 2 common squares of sides $n^2$ length?" 
I'll interpret this as asking, given $n$ distinct symbols, is it possible to create an arbitrarily large grid without having two identical squares of side $n$? 
But of course there are only finitely many distinct order $n$ squares on $n$ symbols, namely, $n^{n^2}$. As soon as your grid is large enough, it will have more than $n^{n^2}$ order $n$ squares, so, by the Pigeonhole Principle, it will have two identical order $n$ squares. Indeed, the number of order $n$ squares in an order $t$ square grid is $(t-n+1)^2$, so as soon as $t\gt n^{n^2/2}+n-1$, there will be a repeated order $n$ square. 
For example, for $n=2$, any $6\times6$ grid must have a repeated $2\times2$ square. I wonder whether a $5\times5$ grid can avoid a repeated $2\times2$. There are 16 different $2\times2$ squares on 2 symbols, and a $5\times5$ grid has 16 $2\times2$ squares, so there's enough room for them all to be different --- but is it actually possible?
