Three. (25 points) Select $k$ numbers from 1 to 2012, such that among the selected $k$ numbers, there are definitely three numbers that can form the lengths of the sides of a triangle (the lengths of the three sides of the triangle must be distinct). What is the minimum value of $k$ that satisfies the condition?
Three, the problem is equivalent to:
Select $k-1$ numbers from $1,2, \cdots, 2012$, such that no three of these numbers can form the sides of a triangle with unequal sides. What is the maximum value of $k$ that satisfies this condition?
Now consider the arrays that meet the above conditions.
When $k=4$, the smallest three numbers are $1, 2, 3$. This array can be expanded continuously, as long as the added number is greater than or equal to the sum of the two largest numbers in the existing array. To maximize $k$, the added number should equal the sum of the two largest numbers in the existing array. Thus, we get
\[
\begin{array}{l}
1,2,3,5,8,13,21,34,55,89,144, \\
233,377,610,987,1597,
\end{array}
\]
a total of 16 numbers.

For any array $a_{1}, a_{2}, \cdots, a_{n}$ that meets the above conditions, it is clear that $a_{i}$ is always greater than or equal to the $i$-th number in (1).
Therefore, $n \leqslant 16 \leqslant k-1$.
Thus, the minimum value of $k$ is 17.
(Chang Yongsheng, Nankai High School, Tianjin, 300100)