Example 5 There are 18 teams participating in a single round-robin tournament, meaning each round the 18 teams are divided into 9 groups, with each group's 2 teams playing one match. In the next round, the teams are regrouped to play, for a total of 17 rounds, ensuring that each team plays one match against each of the other 17 teams. After $n$ rounds of the tournament according to any feasible schedule, there always exist 4 teams that have played a total of only 1 match among themselves. Find the maximum possible value of $n$. (2002 China Mathematical Olympiad Problem)
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        Solution: Consider the following match program:
        1. (1, 2) (3, 4) (5, 6) (7, 8) (9, 18) (10, 11) (12, 13) (14, 15)
(16, 17);
        2. (1, 3) (2, 4) (5, 7) (6, 9) (8, 17) (10, 12) (11, 13) (14, 16)
(15, 18);
        3. (1, 4) (2, 5) (3, 6) (8, 9) (7, 16) (10, 13) (11, 14) (12, 15)
(17, 18);
        4. (1, 5) (2, 7) (3, 8) (4, 9) (6, 15) (10, 14) (11, 16) (12, 17)
(13, 18);
        5. (1, 6) (2, 8) (3, 9) (4, 7) (5, 14) (10, 15) (11, 17) (12, 18)
(13, 16);
        6. (1, 7) (2, 9) (3, 5) (6, 8) (4, 13) (10, 16) (11, 18) (12, 14)
(15, 17);
        7. (1, 8) (2, 6) (4, 5) (7, 9) (3, 12) (10, 17) (11, 15) (13, 14)
(16, 18);
        8. (1, 9) (3, 7) (4, 6) (5, 8) (2, 11) (10, 18) (12, 16) (13, 15)
(14, 17);
        9. (1, 10) (2, 3) (4, 8) (5, 9) (6, 7) (11, 12) (13, 17) (14, 18)
(15, 16);
        10. (1, 11) (2, 12) (3, 13) (4, 14) (5, 15) (6, 16) (7, 17) (8, 18)
(9, 10);
        11. (1, 12) (2, 13) (3, 14) (4, 15) (5, 16) (6, 17) (7, 18) (8, 10)
(9, 11);
        12. (1, 13) (2, 14) (3, 15) (4, 16) (5, 17) (6, 18) (7, 10) (8, 11)
(9, 12);
        *.
        17. (1, 18) (2, 10) (3, 11) (4, 12) (5, 13) (6, 14) (7, 15) (8, 16)
(9, 17).
```
The first 9 teams are called Group A, and the last 9 teams are called Group B. It is easy to see that after 9 rounds, any two teams in the same group have already played against each other. Therefore, any 4 teams have already played at least 2 matches, which does not meet the requirements of the problem. If the above program is reversed and the matches are played in the new order, then after 8 rounds, any two teams in the same group have not played against each other. Each team has played one match against 8 of the 9 teams in the other group. At this point, any 4 teams in the same group have not played against each other, while any 4 teams not all in the same group have played at least 2 matches, which does not meet the requirements of the problem. Therefore, \( n \leqslant 7 \). When \( n = 7 \), assume that no 4 teams meet the requirements of the problem. Select 2 teams \( A_{1} \) and \( A_{2} \) that have already played against each other. Each team has played against 6 other teams, and the 2 teams have played against at most 12 other teams. Therefore, there are at least 4 teams \( B_{1} \), \( B_{2} \), \( B_{3} \), and \( B_{4} \) that have not played against \( A_{1} \) and \( A_{2} \). Consider the 4 teams \( A_{1} \), \( A_{2} \), \( B_{i} \), and \( B_{j} \) (1 \leqslant i < j \leqslant 4). According to the assumption, they have played at least 2 matches, so for any 1 \leqslant i < j \leqslant 4, \( B_{i} \) and \( B_{j} \) have played against each other. Since \( B_{1} \) and \( B_{2} \) have played 3 matches in \(\{B_{1}, B_{2}, B_{3}, B_{4}\}\), \( B_{1} \) and \( B_{2} \) have played 1 match against 4 of the 14 other teams, so there are at least 6 teams \( C_{1} \), \( C_{2} \), \( C_{3} \), \( C_{4} \), \( C_{5} \), and \( C_{6} \) that have not played against \( B_{1} \) and \( B_{2} \). Similarly, for any 1 \leqslant i < j \leqslant 6, \( C_{i} \) and \( C_{j} \) have played against each other. Since \( C_{1} \) and \( C_{2} \) have played 5 matches in \(\{C_{1}, C_{2}, \cdots, C_{6}\}\), \( C_{1} \) and \( C_{2} \) have played 1 match against 2 of the 12 other teams, so there are at least 8 teams \( D_{1} \), \( D_{2} \), \(\cdots\), \( D_{8} \) that have not played against \( C_{1} \) and \( C_{2} \). Similarly, for any 1 \leqslant i < j \leqslant 8, \( D_{i} \) and \( D_{j} \) have played against each other. Thus, \( D_{1} \) and \( D_{2} \) have not played against 10 other teams. Since only 7 rounds have been played, at least 2 teams \( E_{1} \) and \( E_{2} \) among the 10 other teams have not played against each other, so \( D_{1} \), \( D_{2} \), \( E_{1} \), and \( E_{2} \) have played only 1 match, which contradicts the assumption. In summary, the maximum possible value of \( n \) is 7.
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