TITLE: Time to generate a filled-in sub-checkered board
QUESTION [5 upvotes]: Take an $m \times n$ checkered board and one-at-a-time add a piece to an empty square.  At what point are you guaranteed to have an $s \times t$ sub-board where all of its squares are filled?
Here I don't demand that the $s$ rows or $t$ columns be adjacent.  Any selection of $s$ rows and $t$ creates a sub-board -- namely, the $s \cdot t$ intersection squares.
Thanks!

REPLY [2 votes]: This is likely useless,
but for $n{=}m{=}8$, an $8 \times 8$ chessboard,
and $s{=}t{=}2$, a $2 \times 2$ subboard,
with $k$ pieces added randomly (the horizontal axis),
the probability (vertical axis) of a filled $2 \times 2$ 
subboard occuring somewhere on the $8 \times 8$ chessboard,
in $100$ random trials, was:

         


By $k{=}21$, the observed frequency is $97$%.
More precisely, here are the counts out of $100$:
$$
\left(
\begin{array}{cccccccccccccccc
   cccccccc}
 5 & 6 & 7 & 8
   & 9 & 10 & 11 & 12 & 13 &
   14 & 15 & 16 & 17 & 18 & 19
   & 20 & 21 & 22 & 23 & 24 \\
 1 & 2 & 2 & 8
   & 12 & 22 & 21 & 38 & 43 &
   57 & 63 & 68 & 84 & 91 & 95
   & 95 & 97 & 98 & 98 & 99 \\
\end{array}
\right)
\;.
$$