A well-known puzzle goes:

"Suppose that you have 25 horses and a racetrack on which you can race up to 5 horses. If the outcome of each race only tells you the relative speeds of the horses in the race, how many races do you need to determine the fastest 3 horses (and what is the strategy)?"

The solution (look away now if you don't want a spoiler) is to arrange the horses into groups of five and race them, labeling the horses $a_1,\dots,a_5$, ..., $e_1,\dots,e_5$ -- for example, the horse in position 3 in the second race gets the label $b_3$.

Then race horses $a_1, b_1, c_1, d_1, e_1$, and relabel the horses so that all those in the same group as the winner of this race get the label $a_j, j=1,\dots,5$ and so on. Finally, race horses $a_2, a_3, b_1, b_2, c_1$ -- the three fastest horses are now $a_1$ and the two fastest from the final race.

The question: Does this strategy generalize to $m$ horses and $n$ tracks where you want to find the fastest $k$ horses?