For example, 510 people go to the bookstore to buy books, it is known: (1) each person bought three kinds of books; (2) any two people have at least one book in common. How many people at most bought the book that was purchased by the least number of people?
(1993 China Mathematical Olympiad Problem)
Let the number of people who bought the most popular book be $x$. Among the 10 people, person A bought three books. Since the other 9 people all have at least one book in common with A, and $9 \div 3=3$, it follows that among A's three books, the most popular one must have been bought by at least 4 people, so $x \geqslant 4$.
If $x=4$, then each of A's three books was bought by 4 people.
Similarly, each book bought by the other 9 people was also bought by 4 people.
Therefore, the total number of books bought by the 10 people should be a multiple of 4, i.e., 4 | 30, which is a contradiction. Therefore, $x \geqslant 5$.
When $x=5$, let $A_{i}$ represent different books. The scenario where the most popular book is bought by exactly 5 people can be represented by the following sets: $\left\{A_{1}, A_{2}, A_{3}\right\},\left\{A_{1}, A_{2}, A_{6}\right\},\left\{A_{2}, A_{3}, A_{4}\right\},\left\{A_{1}, A_{4}, A_{6}\right\},\left\{A_{1}, A_{4}, A_{5}\right\}$, $\left\{A_{2}, A_{4}, A_{5}\right\},\left\{A_{1}, A_{3}, A_{5}\right\},\left\{A_{2}, A_{5}, A_{6}\right\},\left\{A_{3}, A_{5}, A_{6}\right\},\left\{A_{3}, A_{4}, A_{6}\right\}$.