5-116 Suppose among $r$ people, the transmission of messages is carried out through telephone calls. When two people $A$ and $B$ are talking on the phone, $A$ tells $B$ everything he knows at that moment, and $B$ also tells $A$ everything he knows at that moment. Let $a_{r}$ denote the minimum number of phone calls required among these $r$ people so that everyone knows all the messages of the $r$ people.
(1) Find $a_{3}$;
(2) Find $a_{4}$;
(3) For $r \geqslant 3$, prove that $a_{r} \leqslant a_{r-1}+2$.
[Solution] Let the $r$ people be denoted as $A_{1}, A_{2}, \cdots, A_{r}$.
(1) When $r=3$, first $A_{1}$ and $A_{2}$ communicate, then $A_{2}$ and $A_{3}$ communicate. At this point, $A_{2}$ and $A_{3}$ know all the messages of the 3 people, but $A_{1}$ has not yet learned $A_{3}$'s message, so $A_{1}$ still needs to communicate with $A_{3}$ or $A_{2}$ one more time. It is clear that $a_{3}=3$.
(2) When $r=4$, 4 communications are sufficient: $\left\{A_{1}, A_{2}\right\},\left\{A_{3}, A_{4}\right\},\left\{A_{1}, A_{4}\right\},\left\{A_{2}, A_{3}\right\}$. This satisfies the requirement, so $a_{4} \leqslant 4$.

If only 3 communications are made, then among the 4 people, at least two people each communicate only once (if someone does not communicate at all, it obviously does not meet the requirement). Then these two people, if they communicate once, will not know the messages of the others. If these two people each communicate with others, the one who communicates first will not know the message of the one who communicates later. This shows that no matter how the 3 communications are made, the requirement cannot be met.
Therefore, $a_{4}=4$.
(3) When $r \geqslant 5$, first let $A_{r}$ and $A_{r-1}$ communicate, then by the induction hypothesis, $A_{1}, A_{2}, \cdots, A_{r-1}$ need to communicate $a_{r-1}$ times to know all the messages of the first $r-1$ people. Finally, $A_{1}$ communicates with $A_{r}$ once, and $A_{r}$ also knows all the messages. This shows that $a_{r} \leqslant a_{r-1}+2$. And for $r=3$, 4, this inequality is obviously also true.