Example 3: 10 people go to a bookstore to buy books. It is known that each person bought three types of books, and any two people have at least one book in common. Question: What is the maximum number of people who bought the most purchased book, at a minimum? (No. 8


Let the total number of books sold be $n$, and the set of books bought by the $i$-th person be $A_{i}(i=1,2, \cdots, 10)$. Construct a table of set element relationships, where the $i$-th row has $m_{i}$ ones.

Estimate the total number of times each element appears, we have
$$\sum_{i=1}^{n} m_{i}=S=\sum_{i=1}^{10}\left|A_{i}\right|=\sum_{i=1}^{10} 3=30$$

Next, calculate the total number of times each element appears in the intersections, we have
$$\sum_{i=1}^{n} \mathrm{C}_{m_{i}}^{2}=\sum_{1 \leqslant i<j \leqslant 10}\left|A_{i} \cap A_{j}\right|$$

Let the maximum of $m_{i}$ be $m$, then
$$\begin{aligned}
90 & =2 \mathrm{C}_{10}^{2}=2 \sum_{1 \leqslant i<j \leqslant 10} 1 \leqslant 2 \sum_{1 \leqslant i<i \leqslant 10}\left|A_{i} \cap A_{j}\right| \\
& =2 \sum_{i=1}^{n} \mathrm{C}_{m_{i}}^{2}=\sum_{i=1}^{n} m_{i}^{2}-\sum_{i=1}^{n} m_{i}=\sum_{i=1}^{n} m_{i}^{2}-30 \\
& \leqslant \sum_{i=1}^{n}\left(m_{i} \cdot m\right)-30=m \sum_{i=1}^{n} m_{i}-30 \\
& =30(m-1)
\end{aligned}$$

Solving this inequality, we get $m \geqslant 4$. If $m=4$, then the inequality (1) holds with equality, so all $m_{i}=4(i=1,2, \cdots, n)$. This gives $4 n=\sum_{i=1}^{n} m_{i}=30$, so $4 \mid 30$, which is a contradiction. Therefore, $m \geqslant 5$.

When $m=5$, by the maximality of $m$, all $m_{i} \leqslant 5$, we have $30=\sum_{i=1}^{n} m_{i} \leqslant 5 n, n \geqslant 6$. We take $n=6$, and get a valid construction as shown in the table below:
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline & $A_{1}$ & $A_{2}$ & $A_{3}$ & $A_{4}$ & $A_{5}$ & $A_{6}$ & $A_{7}$ & $A_{8}$ & $A_{9}$ & $A_{10}$ \\
\hline 1 & $*$ & $*$ & $*$ & $*$ & $*$ & & & & & \\
\hline 2 & $*$ & $*$ & & & & $*$ & $*$ & $*$ & & \\
\hline 3 & $*$ & & $*$ & & & $*$ & & & $*$ & $*$ \\
\hline 4 & & & & $*$ & $*$ & $*$ & $*$ & & $*$ & \\
\hline 5 & $*$ & & $*$ & $*$ & & & $*$ & $*$ & & $*$ \\
\hline 6 & & $*$ & & & $*$ & & & $*$ & $*$ & $*$ \\
\hline
\end{tabular}

Thus, the minimum value of $m$ is 5.