Three. (50 points) Given that $n$ is a positive integer, and the set $I=\{1,2, \cdots, n\}(n>3)$ has $k$ three-element subsets $A_{1}, A_{2}, \cdots, A_{k}$ satisfying: for any other three-element subset $B$ of $I$, there exists an integer $m$ and a subset $A_{i}(1 \leqslant i \leqslant k)$ such that
$$
\left\{y \mid y \equiv x+m(\bmod n), 1 \leqslant y \leqslant n, x \in A_{i}\right\}=B .
$$

Find the minimum value of $k$.
If $a, b, c \in \mathbf{Z}_{+}$, and $a+b+c=p$, then $(a, b, c)$ is called a "cyclic group" of length $p$, and it is agreed that $(a, b, c)$, $(b, c, a)$, and $(c, a, b)$ are the same cyclic group.
Consider the number of cyclic groups of length $n$.
The number of cyclic groups where two of $a, b, c$ are equal is $\left[\frac{n-1}{2}\right]$; if $31 n$, and $a, b, c$ are all distinct, the number of cyclic groups is $\frac{1}{3}\left[\mathrm{C}_{n-1}^{2}-1-3\left(\left[\frac{n-1}{2}\right]-1\right)\right]$;
if $n \neq 0(\bmod 3)$, the number of cyclic groups where $a, b, c$ are all distinct is $\frac{1}{3}\left(\mathrm{C}_{n-1}^{2}-3\left[\frac{n-1}{2}\right]\right)$.
In summary, the total number of different cyclic groups of length $n$ is $1+\left[\frac{n^{2}-3 n}{6}\right]$.
For each cyclic group $(a, b, c)$ of length $n$, take a three-element subset $B=\{1, a+1, a+b+1\}$ of set $I$, and there exists a subset $A_{i}$ corresponding to it, and it is easy to verify that different cyclic groups correspond to different subsets $A_{i}$, thus, $k \geqslant 1+\left[\frac{n^{2}-3 n}{6}\right]$.

On the other hand, for each of the first $1+\left[\frac{n^{2}-3 n}{6}\right]$ cyclic groups $(a, b, c)$, take the corresponding subset $\{1,1+a, 1+a+b\}$, obtaining $1+\left[\frac{n^{2}-3 n}{6}\right]$ different subsets.
Next, we show that these subsets meet the requirements.
In fact, for each subset $B=\left\{x_{1}, x_{2}, x_{3}\right\}$ of set $I$ (assuming $x_{1}<x_{2}<x_{3}$), let
$$
a=x_{2}-x_{1}, b=x_{3}-x_{2}, c=n-\left(x_{3}-x_{1}\right) .
$$

Then we get a cyclic group $(a, b, c)$ of length $n$, and the subset $A_{i}$ corresponding to this cyclic group satisfies the existence of an integer $m=x_{1}-1$ (or $x_{2}-1$ or $x_{3}-1$) such that
$$
\begin{array}{l}
\left\{y \mid y \equiv x+m(\bmod n), 1 \leqslant y \leqslant n, x \in A_{i}\right\}=B . \\
\text { In summary, } k_{\min }=1+\left[\frac{n^{2}-3 n}{6}\right] .
\end{array}
$$