6. Let $n$ students be such that among any 3 of them, there are two who know each other, and among any 4 of them, there are two who do not know each other. Find the maximum value of $n$.
(5th China Western Mathematical Olympiad)
6. The maximum value of $n$ sought is 8. When $n=8$, the example shown in the figure satisfies the requirements, where $A_{1}$, $A_{2}, \cdots, A_{8}$ represent 8 students, and a line between $A_{i}$ and $A_{j}$ indicates that $A_{i}$ and $A_{j}$ know each other, otherwise they do not. Now, assuming $n$ students meet the requirements, we will prove that $n \leqslant 8$. For this, we first prove that the following two scenarios are impossible: (1) If a person $A$ knows at least 6 people, denoted as $B_{1}, \cdots, B_{6}$, by Ramsey's theorem, among these 6 people, there exist 3 who do not know each other (which contradicts the given condition that any 3 people have 2 who know each other); or there exist 3 people who all know each other, in which case $A$ and these 3 people form a group of 4 who all know each other, also contradicting the given condition. (2) If a person $A$ knows at most $n-5$ people, then there are at least 4 people who do not know $A$, thus these 4 people must all know each other, which is a contradiction. Secondly, when $n \geqslant 10$, either (1) or (2) must occur, so $n$ does not meet the requirements at this time; when $n=9$, to make (1) and (2) not occur, each person must know exactly 5 other people, in which case the number of friend pairs (pairs who know each other) generated by the 9 people is $\frac{9 \times 5}{2} \notin \mathbf{N}_{+}$, which is a contradiction! From the above, we know that the only $n$ that meets the requirements is $n \leqslant 8$. In summary, the maximum value of $n$ sought is 8.