Let's say I have 10 teams competing in a round-robin tournament. Ties are not allowed, and only teams that have at least 7 wins can move on to a new tournament. How would I go about proving the maximum number of teams that could reach 7 wins and move on?
I don't really know too much about combinatories, but I found a solution!
45 games are played in total, so I would go this way to get some upper bounds:
We first assume that 7 teams qualify, but this would require the sum of wins from those 7 teams to be at least 49, which cannot happen.
Therefore, the soutuion has to be 6 or less, but 6 is also out of the question, since the other 4 teams play 6 games among themselves, so the sum of the top 6 teams has to be 39 or less.
So we are left with 5 at best, where 5 teams can get at most 35 wins, and... there is a concrete solution!
Teams A,B,C,D and E beat every game against teams F,G,H,I and J.
A beats B and C,
B beats C and D,
C beats D and E,
D beats A and E,
E beats A and B
Teams A,B,C,D and E have 7 wins each!
I am not exactly sure how this reasoning generalizes, though!