34. A company gathered for a meeting. Let's call a person sociable if in this company they have at least 20 acquaintances, and at least two of them are acquainted with each other. Let's call a person shy if in this company they have at least 20 strangers, and at least two of them are strangers to each other. It turned out that in the gathered company there are neither sociable nor shy people. What is the maximum number of people that can be in it

$\left(\right.$ Folklore $^{(2)}$
34. Answer: 40.

Evaluation: Suppose that a certain person (let's call him Kostya) is acquainted with at least 20 people. If some of Kostya's acquaintances know each other, then Kostya is sociable. Let's consider the case where all of Kostya's acquaintances do not know each other. In this case, if Kostya is acquainted with at least 21 people, then any of his acquaintances are shy.

Similarly, if Kostya is not acquainted with 20 people, among whom there are some who do not know each other, then Kostya is shy. And in the case where Kostya is not acquainted with 21 people, and all those who are not acquainted with Kostya know each other, then all of them are sociable.

This means that if there are at least 41 people in the company, then Kostya is acquainted with exactly 20 people and is not acquainted with exactly 20 people. In this case, all of Kostya's acquaintances do not know each other, and all those who are not acquainted with Kostya know each other. Consider any person acquainted with Kostya, Vasya, and any person not acquainted with Kostya, Petya. If Vasya and Petya know each other, then Petya is sociable, as he knows Vasya and 19 other people who are not acquainted with Kostya, while those not acquainted with Kostya know each other. Similarly, if Vasya and Petya do not know each other, then Vasya is shy.

Example: Let's divide a company of 40 people into two groups of 20 people each. Suppose that people in one group know each other, while people from different groups do not know each other. Then, if there were a sociable person, all his acquaintances would be in the same group with him, and then there would be at least 21 people in this group. If a shy person were found, then all people not acquainted with him would be in the same group, and thus would know each other.