For a positive integer $k$, let the set
$$
R_{n}=\left\{\begin{array}{l}
\{-k,-(k-1), \cdots,-1,1, \cdots, k-1, k\}, \quad n=2 k ; \\
\{-k,-(k-1), \cdots,-1,0,1, \cdots, k-1, k\}, \quad n=2 k+1 .
\end{array}\right.
$$

A device contains some balls and red or white strings connecting some pairs of balls. A "labeling" is to color each ball with one of the elements from $R_{n}$. A labeling is "good" if and only if the colors of any two balls connected by a string are different; a labeling is "wise" if and only if the colors of any two balls connected by a white string are different, and the sum of the colors of any two balls connected by a red string is not equal to 0.

For a fixed positive integer $n \geqslant 3$, assume that any device has a good labeling with respect to $R_{n}$, and also has a wise labeling with respect to $R_{m}$. Find the minimum value of the positive integer $m=m(n)$.
6. $m=2 n-1$.

First, prove: If a device has a good labeling with respect to $R_{n}$, then it also has an insightful labeling with respect to $R_{m} (m=2 n-1)$.
In fact, in the set
$$
\begin{array}{l} 
R_{m}=R_{2 n-1} \\
=\{-(n-1),-(n-2), \cdots,-1,0,1, \cdots, \\
n-2, n-1\}
\end{array}
$$

there are $n$ non-negative integers, then any good labeling of the device with respect to these $n$ non-negative integers is an insightful labeling.

Next, construct a device that has a good labeling with respect to $R_{n}$, and for any positive integer $m<2 n-1$, it has no insightful labeling.

Consider a $(2 n-1) \times n$ rectangular grid, with a ball placed in each cell. Connect any two balls in the same row with a red string, and connect any two balls in different rows and different columns with a white string (no strings connect any two balls in the same column). Then this device has a good labeling with respect to $R_{n}$.

In fact, since there are $n$ different colors, if all balls in the same column are colored the same color, and balls in different columns are colored different colors, then a good labeling is obtained. Assume this device has an insightful labeling with respect to $R_{m}$. Below, we discuss two cases.
(1) There exists a row where all balls have different colors.
Since any two balls in this row are connected by a red string, for each integer $k$, the colors $-k$ and $k$ are used at most once. Therefore, the distinct absolute values of all elements in $R_{m}$ are at least $n$.
Thus, $m \geqslant 2 n-1$.
(2) Each row contains two balls of the same color.
Assume the two colors of the balls in the same row in some two rows are $a$ and $b$.

Since any two balls in different rows and different columns are connected by a white string, $a \neq b$.

Thus, the colors of the balls in any two rows that contain two balls of the same color are different. This indicates that the number of different colors is at least $2 n-1$.
Thus, $m \geqslant 2 n-1$.