4.73 A city has purchased a batch of public buses, planning to establish 1983 bus stops and set up several routes between them, so that
（1）as many routes as possible can be set up;
（2）any two routes share at least one bus stop;
(3) each bus stop is on at most two different routes.

How many routes can be set up at most under the above requirements? How many bus stops should each route have at least?
[Solution] Let $S$ denote the set of 1983 stations. Suppose a total of $k$ lines are set up, and let $A_{1}, A_{2}, \cdots, A_{k}$ represent the sets of all stations on the $1, 2, \cdots, k$ lines, respectively. Thus, according to the given conditions, we have:
(i) For any $1 \leqslant i<j \leqslant k, A_{i} \cap A_{j} \neq \varnothing$;
(ii) For any $1 \leqslant i<j<m \leqslant k, A_{i} \cap A_{j} \cap A_{m}=\varnothing$. From (i), we know that each set $A_{i}$ shares at least one common element with each of the other $k-1$ sets. From (ii), we know that these common elements are distinct. Therefore, $\left|A_{i}\right| \geqslant k-1$, for $i=1, \cdots, k$. Thus, the total number of elements in the sets $A_{1}, A_{2}, \cdots, A_{k}$ is at least $k(k-1)$. In this counting process, each element is counted twice, so the set $A_{1} \cup A_{2} \cup \cdots \cup A_{k}$ has at least $\frac{1}{2} k(k-1)$ elements. Hence, we have
$$
\frac{1}{2} k(k-1) \leqslant 1983 \text {. }
$$

Solving this, we get $k \leqslant 63$.
It is easy to see that setting up 63 lines is feasible. Therefore, the maximum number of lines that can be set up is 63, and each line should pass through at least 62 stations.