Anybody?

Hint: try to use Turan's theorem

my solutioonn :n$\leq 9$

can someone post a full solution?

This is asking for the value of the ramsey number R(3,4) it is 9 so the maximum wanted is 8. You can search the web for proof of this.

we will prove that $ 8\ge n$ (*)if someone A knows at least 6 people, by Ramsey's theorem, there exist 3 people among them who dont know each other in every three students, or there exist 3 people they know each other. A and the three people form a group of four people such that every two of them know each other. contradiction (**)if some one A knows at most n-5 people, then in the remaining there are at least 4 people, none of knows A. Thus every two of the four people know each other, contradiction. when $ n\ge10$ one of (*) and (**) must occur if $ n = 9$ in order to avoid (*) and (**), each person knows exactly 5 pther people. Thus the number of pairs knowing each other make a contradiction.This contradiction implies $ 8\ge n$.Hence, the maximum value is 8.

Here is the picture for $n=8$ (my sol is the same as aboves): Here, where $A_1 , A_2 , \dots, A_8$ represent the $8$ students. The line segment between $A_i$ and $A_j$ means $A_i$ and $A_j$ know each other.