6. Students $M_{1}, M_{2}, M_{3}, M_{4}$ go to buy 10 different books numbered $1, 2, 3, \cdots, 10$. To save money and facilitate sharing, they agree that each person will only buy 5 of these books, and no two students can buy all 10 books, while any three students must buy all 10 books. When $M_{1}$ buys books numbered $1, 2, 3, 4, 5$, $M_{2}$ buys books numbered $5, 6, 7, 8, 9$, and $M_{3}$ buys books numbered $1, 2, 3, 9, 10$, to meet the above requirements, the books that $M_{4}$ should buy are $\qquad$
6. $4,6,7,8,10$.

Let the set of book numbers bought by $M_{i}$ be $A_{i}, i=1,2,3,4$. Let the universal set $U=\{1,2,3, \cdots, 10\}$.
Since $A_{1}=\{1,2,3,4,5\}, A_{2}=\{5,6,7,8,9\}$,
$$
\begin{array}{l}
A_{3}=\{1,2,3,9,10\}, \overline{A_{1} \cup A_{2}}=\{10\}, \\
\overline{A_{2} \cup A_{3}}=\{4\}, \overline{A_{3} \cup A_{1}}=\{6,7,8\},
\end{array}
$$

We have $A_{4} \supseteq\left(\overline{A_{1} \cup A_{2}}\right) \cup\left(\overline{A_{2} \cup A_{3}}\right) \cup\left(\overline{A_{3} \cup A_{1}}\right)$
$$
=\{10\} \cup\{4\} \cup\{6,7,8\}=\{4,6,7,8,10\} \text {. }
$$

Also, $\left|A_{4}\right|=5$, so $A_{4}=\{4,6,7,8,10\}$.