I came up with a question several hours ago...but I couldn't find any information about it.
The problem goes like below
$$P^n_k =\{(p_1,...,p_n)|p_1<...<p_n:primes,p_n-p_1\le k\}$$
$$k_n=min\{k|card(P^n_k)=\infty\}$$
Is $k_n=O(n^m)$ for some $m\in \mathbb N$?
If not, how about log or exponential functions?
I do not expect to get the answer. However, I found this problem interesting and wish to discuss with others about it. I also look forward to some knowledge that can help me to attack this problem.
Thanks for helping~
Edit. I noticed that we should prove the existence of $k_n$ first.