lets play tennis!

In a tournament with $8$ people, since everyone plays against everyone else, there will be $\displaystyle\binom{8}{2} = 28$ games. Therefore, the total number of victories in the tournament is $28$ . Therefore, the average number of victories per player is $\dfrac{28}{8} = 3.5$ . Therefore, by the Pigeonhole Principle, there must be a least one player who won at least $4$ games. Let's call this player $W$ , and we'll call the four losers $A, B, C, D$ .

Now since there are $4$ of them, the players $A, B, C, D$ played $\displaystyle\binom{4}{2} = 6$ games amongst each other. Hence the average number of losses in this group is $\dfrac{6}{4} = 1.5$ . Therefore, by the Pigeonhole Principle, there is at least one player (let's say $D$ ) who lost at least two games to other players in the group. Say $D$ lost to $A$ and $B$ . Then the set ${W, A, B, D}$ is ordered with the clear winner $W$ and the clear loser $D$ .

Therefore, any tournament of $8$ people has an ordered group.

However we're not done yet; now let's show that not all tournaments of $7$ players have an ordered group. Consider the following tournament with $7$ players numbered $1$ through $7$ : $\left\{\begin{matrix} Player& 1& beats& 2,3,4 \\ Player& 2& beats& 3,5,7 \\ Player& 3& beats& 4,5,6 \\ Player& 4& beats &2,6,7 \\ Player&5 &beats& 1,4,7 \\ Player& 6& beats& 1,2,5 \\ Player& 7& beats& 1,3,6 \end{matrix}\right.$ Each of the players here beats $3$ players and loses to $3$ players. Notice that each of the above triplets creates a "triangle" in which each player wins exactly one game and loses exactly one game amongst each other. For example, in the first row we have $1$ beats $2,3$ and $4$ , where $2$ beats $3,$ $3$ beats $4$ and $4$ beats $2$ .

Now in order to have an "ordered group", we must have one "clear winner" and one "clear loser". If there were a clear winner, however, then the three people that he/she won against would create a triangle and thus have no clear loser. Hence, this tournament has no ordered groups.

Therefore, the minimum is $8$ .