6. A class arranges for some students to participate in activities over a week (6 days), with several people arranged each day. However, among any three days, there must be at least one student who participates in all three days. Among any four days, there should not be a student who participates in all four days. How many students are needed at minimum, and provide a specific arrangement plan for the activities according to the conditions.

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。
6. From the given information, since there is no activity that someone attends for four days, each person can attend activities for at most three days, with $n$ students.

And there is no student who attends two different sets of three days (not completely overlapping) (otherwise, this person would have attended activities for at least four days, which is a contradiction!).
Furthermore, from the given information, for any fixed three days, the students attending activities must be different from those attending activities on any other three days.
$$
\Rightarrow n \geqslant C_{6}^{3}=20 \text {. }
$$

Below is an example to show that $n=20$ is possible.
The 20 people attend activities on the following days:
$$
\begin{array}{l}
(1,2,3)(1,2,4)(1,2,5)(1,2,6)(1,3,4) \\
(1,3,5)(1,3,6)(1,4,5)(1,4,6)(1,5,6) \\
(2,3,4)(2,3,5)(2,3,6)(2,4,5)(2,4,6) \\
(2,5,6)(3,4,5)(3,4,6)(3,5,6)(4,5,6)
\end{array}
$$