Three, (50 points) Find the maximum value of the positive integer $r$ such that: for any five 500-element subsets of the set $\{1,2, \cdots, 1000\}$, there exist two subsets that have at least $r$ elements in common.

Find the maximum value of the positive integer $r$ such that: for any five 500-element subsets of the set $\{1,2, \cdots, 1000\}$, there exist two subsets that have at least $r$ elements in common.
Three, first explain $r \leqslant 200$.
Take $k \in\{1,2, \cdots, 10\}$. Let
$$
A_{k}=\{100 k-99,100 k-98, \cdots, 100 k\} \text {. }
$$

Consider the set
$$
\begin{array}{l}
A_{1} \cup A_{5} \cup A_{6} \cup A_{7} \cup A_{9}, A_{1} \cup A_{2} \cup A_{7} \cup A_{8} \cup A_{10}, \\
A_{2} \cup A_{3} \cup A_{6} \cup A_{8} \cup A_{9}, A_{3} \cup A_{4} \cup A_{7} \cup A_{9} \cup A_{10}, \\
A_{4} \cup A_{5} \cup A_{6} \cup A_{8} \cup A_{10},
\end{array}
$$

It can be seen that they satisfy the problem's conditions and each set has 200 elements. Thus, $r \leqslant 200$.
Define $a_{i j}=\left\{\begin{array}{l}1, i \in A_{j} ; \\ 0,\end{array} m_{i}=\sum_{j=1}^{5} a_{i j}\right.$,

where $i=1,2, \cdots, 1000 ; j=1,2, \cdots, 5$.
Then $\sum_{i=1}^{1000} m_{i}=2500$.
$$
\begin{array}{l}
\text { By } \sum_{1 \leq i<j \leq 5}\left|A_{i} \cap A_{j}\right|=\sum_{i=1}^{1000} \mathrm{C}_{m_{i}}^{2} \\
=\frac{1}{2}\left(\sum_{i=1}^{1000} m_{i}^{2}-\sum_{i=1}^{1000} m_{i}\right), \\
\sum_{i=1}^{1000} m_{i}^{2} \geqslant \frac{1}{1000}\left(\sum_{i=1}^{1000} m_{i}\right)^{2},
\end{array}
$$

It is known that $\sum_{i=1}^{1000} m_{i}^{2}$ takes the minimum value when $m_{i}$ are as close as possible.
Suppose there are $x$ 2s and $y$ 3s.
Then $\left\{\begin{array}{l}x+y=1000, \\ 2 x+3 y=2500\end{array} \Rightarrow x=y=500\right.$.
Thus, $\sum_{i=1}^{1000} m_{i}^{2} \geqslant 500 \times 2^{2}+500 \times 3^{2}=6500$
$\Rightarrow \sum_{1 \leqslant i<j<5}\left|A_{i} \cap A_{j}\right| \geqslant 2000$.
Therefore, there must exist $1 \leqslant i<j \leqslant 5$, such that
$\left|A_{i} \cap A_{j}\right| \geqslant \frac{2000}{C_{5}^{2}}=200$.
Thus, $r \geqslant 200$.