6. A company has $n$ employees, and it is known that every two of these $n$ employees can ensure that there are at least three days in a week when one works while the other does not (these days do not have to be the same employee working, and employees can choose not to work the entire week). Find the maximum value of $n$.

将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 

Note: The last sentence is a repetition of the instruction and should not be included in the final translation. Here is the corrected output:

6. A company has $n$ employees, and it is known that every two of these $n$ employees can ensure that there are at least three days in a week when one works while the other does not (these days do not have to be the same employee working, and employees can choose not to work the entire week). Find the maximum value of $n$.
6. The maximum value of $n$ is 16.

To prove $n \leqslant 16$, use a seven-tuple $\left(a_{1}, a_{2}, \cdots, a_{7}\right)\left(a_{i}=0 \text{ or } 1 ; i=1,2, \cdots, 7\right)$ to represent an employee's attendance over seven days, where $a_{i}$ indicates the attendance on the $i$-th day: if the employee works on the $i$-th day, then $a_{i}=1$; otherwise, $a_{i}=0$. Additionally, if two seven-tuples differ in exactly one position, they are called "adjacent."

Let $v_{1}, v_{2}, \cdots, v_{n}$ represent the seven-tuples corresponding to these $n$ employees.

According to the problem, for any two tuples $v_{i}$ and $v_{j} (i \neq j)$, there are at least three positions with different values. Thus, if we consider the adjacent tuples of $v_{i}$ and $v_{j}$, they must differ in at least 3-1-1=1 position, indicating that the adjacent tuples of $v_{i}$ and $v_{j}$ must also be different.

Considering $v_{1}, v_{2}, \cdots, v_{n}$ and their adjacent seven-tuples, there are at least $n+7n=8n$ distinct seven-tuples.
Since there are $2^{7}=128$ seven-tuples in total, we have:
$8n \leqslant 128 \Rightarrow n \leqslant 16$.
For $n=16$, an example is given:
\[
\begin{aligned}
v_{1} & =(0,0,0,0,0,0,0), \\
v_{2} & =(1,1,1,0,0,0,0), \\
v_{3} & =(1,0,0,1,1,0,0), \\
v_{4} & =(1,0,0,0,0,1,1), \\
v_{5} & =(0,1,0,1,0,0,1), \\
v_{6} & =(0,1,0,0,1,1,0), \\
v_{7} & =(0,0,1,1,0,1,0), \\
v_{8} & =(0,0,1,0,1,0,1), \\
v_{9} & =(1,1,1,1,1,1,1), \\
v_{10} & =(0,0,0,1,1,1,1), \\
v_{11} & =(0,1,1,0,0,1,1), \\
v_{12} & =(0,1,1,1,1,0,0), \\
v_{13} & =(1,0,1,0,1,1,0), \\
v_{14} & =(1,0,1,1,0,0,1), \\
v_{15} & =(1,1,0,0,1,0,1), \\
v_{16} & =(1,1,0,1,0,1,0).
\end{aligned}
\]

To show that this example meets the requirements, use $d\left(v_{i}, v_{j}\right)$ $(1 \leqslant i<j \leqslant 16)$ to denote the number of different positions in the seven-tuples $v_{i}$ and $v_{j}$. It can be verified that:
(1) For $2 \leqslant j \leqslant 8$, $d\left(v_{1}, v_{j}\right)=3$;
For $10 \leqslant j \leqslant 16$, $d\left(v_{1}, v_{j}\right)=4$; $d\left(v_{1}, v_{9}\right)=7$.
(2) For $2 \leqslant i \leqslant 8$,
When $i<j \leqslant 9$, $d\left(v_{i}, v_{j}\right)=4$;
When $10 \leqslant j \leqslant 16$ and $j \neq i+8$, $d\left(v_{i}, v_{j}\right)=3$;
When $j=i+8$, $d\left(v_{i}, v_{j}\right)=7$.
(3) For $10 \leqslant j \leqslant 16$, $d\left(v_{9}, v_{j}\right)=3$;
For $10 \leqslant i<j \leqslant 16$, $d\left(v_{i}, v_{j}\right)=4$.
Thus, the given example meets the requirements, and the maximum value of $n$ is 16.