There are $25$ horses with different speeds. My goal is to rank all of them, by using only runs with $5$ horses, and taking partial rankings. How many runs do I need, at minimum, to complete my task?
As a partial answer, I know that is possible to determine the first $3$ horses with $7$ runs, and, by a slight generalization of the optimal algorithm used to find the first three, have the complete ranking in $20$ runs.
Is it possible to do better?
What if we have $n$ horses and want to rank them with runs with $k$ horses?