Three. (50 points) There are 2008 students participating in a large public welfare activity. If two students know each other, then these two students are considered as a cooperative group.
(1) Find the minimum number of cooperative groups $m$, such that no matter how the students know each other, there exist three students who are pairwise in a cooperative group;
(2) If the number of cooperative groups is $\left[\frac{m}{22}\right]$, prove: there exist four students $A$, $B$, $C$, $D$, such that $A$ and $B$, $B$ and $C$, $C$ and $D$, $D$ and $A$ are each in a cooperative group.
(1) Let $n=1004$.
The following proof: $m=n^{2}+1$.
Divide the students into two large groups, each with $n$ students, and no students in the same large group know each other, while each student knows every student in the other large group, thus forming $n^{2}$ cooperative groups, but there do not exist three students who are all in the same cooperative group.

If there are $n^{2}+1$ cooperative groups, let student $X$ know the most students, and let them be $Y_{1}$, $Y_{2}, \cdots, Y_{k}$.

If there exist $i, j \in\{1,2, \cdots, k\}(i \neq j)$ such that $Y_{i}$ and $Y_{j}$ know each other, then $X, Y_{i}, Y_{j}$ satisfy the condition;

If $Y_{1}, Y_{2}, \cdots, Y_{k}$ do not form any cooperative group, then the number of cooperative groups does not exceed
$$
k+(2 n-k-1) k=k(2 n-k) \leqslant n^{2},
$$

which is a contradiction.
Therefore, $m=1004^{2}+1=1008017$.
(2) Let $n=1004, l=\left[\frac{m}{22}\right]=45818$, and the $2 n$ students are $X_{1}, X_{2}, \cdots, X_{2 n}$, with the number of students they know being $a_{1}, a_{2}, \cdots, a_{2 n}$, then
$$
a_{1}+a_{2}+\cdots+a_{2 n}=2 l \text {. }
$$

Consider all possible pairs of groups among the students each student knows, the total number is
$$
\begin{array}{l}
\mathrm{C}_{a_{1}}^{2}+\mathrm{C}_{a_{2}}^{2}+\cdots+\mathrm{C}_{a_{2 n}}^{2} \\
=\frac{1}{2}\left(\sum_{i=1}^{2 n} a_{i}^{2}-\sum_{i=1}^{2 n} a_{i}\right) \\
\geqslant \frac{1}{2}\left[\frac{1}{2 n}\left(\sum_{i=1}^{2 n} a_{i}\right)^{2}-\sum_{i=1}^{2 n} a_{i}\right] \\
=\frac{1}{2} \sum_{i=1}^{2 n} a_{i}\left(\frac{1}{2 n} \sum_{i=1}^{2 n} a_{i}-1\right) \\
=\frac{2 l}{2}\left(\frac{2 l}{2 n}-1\right)=l\left(\frac{l}{n}-1\right) \\
=45818\left(\frac{45818}{1004}-1\right) \\
>C_{2088}^{2} .
\end{array}
$$

Therefore, there exists a pair of students $A$ and $C$ who both know $B$ and $D$.
(Li Jianquan, Institute of Mathematics Education and Mathematical Olympiad, Tianjin Normal University, 300387)