China Western Mathematical Olympiad

The maximum value of $n$ is $8$. When $n=8$, the example shown in the diagram satisfies the requirements, where $A_1,A_2,\dots ,A_8$ represent $8$ students. The line segment between $A_i$ and $A_j$ means $A_i$ and $A_j$ know each other.

Next, if $n$ students satisfy the conditions, we want to show that $n\le 8$. To do this, we first prove that the following two cases are impossible.

(1) If someone $A$ knows at least $6$ persons, denoted by $B_1,B_2,\dots ,B_6$. By Ramsey's theorem, there exist $3$ persons among them who do not know each other. This contradicts that there are two who know each other in every three students, or that there exist $3$ people they know each other. $A$ and the three persons form a group of four persons such that every two of them know each other. A contradiction.

(2) If some one $A$ knows at most $n-5$ persons, then in the remaining there are at least $4$ persons, none of whom knows $A$. Thus every two of the four persons know each other, a contradiction.

When $n\ge 10$, one of (1) and (2) must occur. So such $n$ does not satisfy the requirements.

If $n=9$, in order to avoid (1) and (2), each person knows exactly $5$ other persons. Thus the number of pairs knowing each other is $\frac{9\times 5}{2}\notin \mathbb{N}$. This contradiction implies $n\le 8$. Thus the maximum value of $n$ is $8$.