16 In four-dimensional space, the distance between point $A\left(a_{1}, a_{2}, a_{3}, a_{4}\right)$ and point $B\left(b_{1}, b_{2}, b_{3}, b_{4}\right)$ is defined as $A B=\sqrt{\sum_{i=1}^{4}\left(a_{i}-b_{i}\right)^{2}}$. Consider the set of points
$$
I=\left\{P\left(c_{1}, c_{2}, c_{3}, c_{4}\right) \mid c_{i}=0 \text { or } 1, i=1,2,3,4\right\},
$$

If for any $n$-element subset $Q=\left\{P_{1}, P_{2}, \cdots, P_{n}\right\}$ of $I$, there exist $P_{i}, P_{j}, P_{k} \in Q$ such that $\triangle P_{i} P_{j} P_{k}$ is an equilateral triangle, i.e., $P_{i} P_{j}=P_{j} P_{k}=P_{k} P_{i}$, find the minimum value of $n$.
(16) Construct the following 8 points: $P_{1}(0,0,0,0), P_{2}(0,1,0,0), P_{3}(0,0,0,1), P_{4}(0,0,1,1), P_{5}(1,1,0,0), P_{6}(1,1,1,0), P_{7}(1,1,1,1), P_{8}(1,0,1,1)$. By calculation, it is known that no three of these points can form an equilateral triangle, so $n_{\text {min }} \geqslant 9$.
When $n=9$, let
$$
S_{m}=\left\{P \mid P \in I \text {, and } P P_{m}=1\right\}(m=1,2, \cdots, 9),
$$

then $\left|S_{m}\right|=4$, so
$$
\sum_{m=1}^{9}\left|S_{m}\right|=4 \times 9=36,
$$

and $|I|=2^{4}=16$, so there exists $P \in I$, such that
$$
P \in S_{i} \cap S_{j} \cap S_{k},
$$

i.e.,
$$
P P_{i}=P P_{j}=P P_{k}=1,
$$

so
$$
P_{i} P_{j}=P_{j} P_{k}=P_{k} P_{i}=\sqrt{2},
$$
$\triangle P_{i} P_{j} P_{k}$ is an equilateral triangle.
In summary, the minimum value of $n$ is 9.