I'm actually looking for a solution of extension of this problem:
We have a set $$M = \{(x,y)\in \mathbb{Z}^2; 0\leq x,y\leq 10\}$$
Let $P$ be a subset of rectangles with vertices in $M$ and with sides parallel to coordinate axis. What is the smallest cardinality of $P$ that we can find for sure in $P$ two rectangles $R$ and $R'$ such that $R\subseteq R'$?
Proof for $551$: Every rectangle is uniquely determined with two parallels with x-axis and two parallel with y-axis. Since we have $11$ parallel lines with x-axis, we can chose those on ${11\choose 2}=55 $ ways. So we have two parallels to x-axis with at least $11$ rectangles with sides on these two parallels.
Now let us observe only these $11$ rectangles. Each is uniquely determined with their left and right side. Since we have $11$ rectangles and only $10$ possibilities we have two of them with left side on the same parallel to y-axis. So one of those includes the other one and we are done.