I am studying MIT OCW lecture notes but they do not have solutions for the following problem.
Directed Acyclic Tournaments
In a round-robin tournament, every two distinct players play against each other just once. For a round-robin tournament with no tied games, a record of who beat whom can be described with a tournament digraph, where the vertices correspond to players and there is an edge $x \rightarrow y$ iff $x$ beat $y$ in their game.
A ranking is a path that includes all the players. So in a ranking, each player won the game against the next lowest ranked player, but may very well have lost their games against much lower ranked players —whoever does the ranking may have a lot of room to play favorites.
- Give an example of a tournament digraph with more than one ranking.
- Prove that if a tournament digraph is a DAG, then it has at most one ranking.
- Prove that every finite tournament digraph has a ranking.
- Prove that the greater-than relation, $>$, on the rational numbers, $Q$, is a DAG and a tournament graph that has no ranking.
I got stuck at questions 2, 3, and 4. I have no idea how to solve it. I tried induction on b without a luck but still induction does not even help for infinite DAG cases.