Example 7 $n$ people greet each other by phone during a holiday, it is known that each person has called at most 3 of their friends; any 2 people have had at most 1 phone call; and among any 3 people, at least 2 of them, one has called the other. Find the maximum value of $n$. (Original problem)
Solve the original problem: Use $n$ points to represent $n$ people. If person $A$ calls person $B$'s home phone, then draw a directed edge from $A$ to $B$, resulting in a simple directed graph $G$.

On one hand, $\bar{G}$ contains no triangles, and by the lemma, $|| \bar{G} || \leqslant \left[\frac{n^{2}}{4}\right]$, so $||G|| = \mathrm{C}_{n}^{2} - || \bar{G} || \geqslant \mathrm{C}_{n}^{2} - \left[\frac{n^{2}}{4}\right] = \left[\frac{(n-1)^{2}}{4}\right]$; on the other hand, $||G|| = \sum_{i=1}^{n} \mathrm{~d}^{+}\left(x_{i}\right) \leqslant \sum_{i=1}^{n} 3 = 3n$, so
$$\left[\frac{(n-1)^{2}}{4}\right] \leqslant 3n$$

When $n$ is odd, (1) becomes $\frac{(n-1)^{2}}{4} \leqslant 3n$, solving this gives $n \leqslant 13$; when $n$ is even, (1) becomes $\frac{n^{2}-2n}{4} \leqslant 3n$, solving this gives $n \leqslant 14$.

In conclusion, $n \leqslant 14$.