Example 2 After a round-robin volleyball tournament, it was found that for any two teams, there is always a third team that has defeated both of these teams. How many teams participated in the tournament at the very least?
Let there be $n$ teams $A_{1}, A_{2}, \cdots, A_{n}$ participating in the competition, and the number of teams they defeated are $d_{1}$, $d_{2}, \cdots, d_{n}$ respectively. Since a total of $C_{n}^{2}$ matches were played, and each match must result in one team being defeated, we have
$$
d_{1}+d_{2}+\cdots+d_{n}=C_{n}^{2} \text {. }
$$

By the given conditions, for any two teams $A_{i}, A_{j}(1 \leqslant i, j \leqslant n)$, there must be a team $A_{k}$ that defeated both $A_{i}$ and $A_{j}(k \neq i, j)$. Similarly, there is a team $A_{t}$ that defeated both $A_{i}$ and $A_{k}(t \neq i, j, k)$, and there is a team $A_{r}$ that defeated both $A_{i}$ and $A_{t}(r \neq t, k, i)$ (in Figure 4-8, $A_{k} \rightarrow A_{j}$ indicates that $A_{k}$ defeated $A_{j}$). Therefore, any team $A_{i}$ is at least defeated by 3 teams $A_{k}, A_{i}, A$, so $d_{i} \leqslant n-4(i=1,2, \cdots, n)$.

If $A_{k}$ defeated both $A_{i}$ and $A_{j}$, then $\left\{A_{k}, A_{i}, A_{j}\right\}$ forms a "trio group". The set of such "trio groups" is denoted as $S$. Since any two teams are at least defeated by one team, forming at least one "trio group", we have
$$
|S| \geqslant C_{n}^{2} \text {. }
$$

On the other hand, for each team $A_{k}$, it defeated $d_{k}$ teams, forming $C_{d_{k}}^{2}$ "trio groups" containing $A_{k}$, so
$$
|S|=\sum_{k=1}^{n} C_{d_{k}}^{2}=\frac{1}{2}\left(\sum_{k=1}^{n} d_{k}^{2}-\sum_{k=1}^{n} d_{k}\right) .
$$

From (1), (2), (3), and (4), we get
$$
n(n-4)^{2} \geqslant \sum_{k=1}^{n} d_{k}^{2} \geqslant 2 C_{n}^{2}+\sum_{k=1}^{n} d_{k}=3 C_{n}^{2} .
$$

That is, $2 n^{2}-19 n+35 \geqslant 0$, or $(2 n-5)(n-7) \geqslant 0$,
so $n \leqslant \frac{5}{2}$ or $n \geqslant 7$.

But since at least 3 teams participate in the competition, we have $n \geqslant 7$.
The table below provides an example of 7 teams participating in the competition that meets the conditions:
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline & $A_{1}$ & $A_{2}$ & $A_{3}$ & $A_{4}$ & $A_{5}$ & $A_{6}$ & $A_{7}$ & Wins & Losses \\
\hline$A_{1}$ & & Loss & Win & Loss & Win & Loss & Win & 3 & 3 \\
\hline$A_{2}$ & Win & & Loss & Win & Loss & Loss & Win & 3 & 3 \\
\hline$A_{3}$ & Loss & Win & & Win & Win & Loss & Loss & 3 & 3 \\
\hline$A_{4}$ & Win & Loss & Loss & & Win & Win & Loss & 3 & 3 \\
\hline$A_{5}$ & Loss & Win & Loss & Loss & & Win & Win & 3 & 3 \\
\hline$A_{6}$ & Win & Win & Win & Loss & Loss & & Loss & 3 & 3 \\
\hline$A_{7}$ & Loss & Loss & Win & Win & Loss & Win & & 3 & 3 \\
\hline
\end{tabular}

Therefore, the minimum number of teams participating is 7.