$7 \cdot 66$ Given a family of sets $\left\{A_{1}, A_{2}, \cdots, A_{n}\right\}$ satisfying the following conditions:
(1) Each set $A_{i}$ contains exactly 30 elements;
(2) For any $1 \leqslant i<j \leqslant n, A_{i} \cap A_{j}$ contains exactly 1 element;
(3) $A_{1} \cap A_{2} \cap \cdots \cap A_{n}=\varnothing$.

Find the maximum value of the number of sets $n$ in this family.
[Solution] Let $A_{1}=\left\{a_{1}, a_{2}, \cdots, a_{30}\right\}$. We point out that each $a_{i}$ belongs to at most 30 sets in this family. Otherwise, suppose $a_{1} \in A_{j}, j=1,2, \cdots, 31$. By (2), the 31 sets
$$
A_{j}-\left\{a_{1}\right\}, j=1,2, \cdots, 31
$$

contain $29 \times 31$ distinct elements. By (3), there exists a set $B$ such that $a_{1} \in B$. Then by (2), $B$ has one common element with each of the 31 sets in (1), so $|B|=$ 31, a contradiction. Thus, each $a_{i}$ belongs to at most 30 sets, in addition to $A_{1}$, there are 29 other sets. Therefore, the number of sets in this family does not exceed $30 \times 29+1=871$.
On the other hand, we construct 871 sets that satisfy the requirements of the problem. Let
$$
\begin{array}{l}
A=\left\{a_{0}, a_{1}, \cdots, a_{29}\right\}, \\
B_{i}=\left\{a_{0}, a_{i 1}, a_{i 2}, \cdots, a_{i 29}\right\}, 1 \leqslant i \leqslant 29, \\
C_{i j}=\left\{a_{i}\right\} \cup\left\{a_{k j+(k-1)(i-1)} \mid k=1,2, \cdots, 29\right\}, \quad 1 \leqslant i, j \leqslant 29,
\end{array}
$$

where different symbols represent different elements, but $a_{k, h}$ and $a_{k, s+29}$ are the same element. These 871 sets satisfy the requirements of the problem.
In fact, we have
$$
\begin{array}{l}
A \cap B_{i}=B_{i} \cap B_{j}=\left\{a_{0}\right\}, 1 \leqslant i<j \leqslant 29, \\
A \cap C_{i j}=\left\{a_{i}\right\}, 1 \leqslant i, j \leqslant 29, \\
B_{s} \cap C_{i j}=\left\{a_{s j+(s-1)(i-1)}\right\}, 1 \leqslant s, i, j \leqslant 29, \\
C_{i j} \cap C_{i k}=\left\{a_{i}\right\}, 1 \leqslant i, j, k \leqslant 29, j \neq k, \\
C_{i j} \cap C_{k j}=\left\{a_{1 j}\right\}, 1 \leqslant i, j, k \leqslant 29, i \neq k, \\
C_{i j} \cap C_{k h}=\left\{a_{s j+(s-1)(i-1)}\right\}, i \neq k, j \neq h,
\end{array}
$$

where the subscript $s$ is the unique solution to the congruence equation $(s-1)(k-i) \equiv j-h(\bmod 29)$.