12. (22 points) If integers $a$ and $b$ are neither coprime nor have a divisibility relationship, then $a$ and $b$ are called a "coalition" pair. Let $A$ be an $n$-element subset of the set $M=\{1,2, \cdots, 2014\}$, and any two numbers in $A$ are coalition pairs. Find the maximum value of $n$.


12. Call such a subset $A$ a "coalition subset".

First, construct a coalition subset with 504 elements. For this, take
$$
A=\{2 k \mid k=504,505, \cdots, 1007\} .
$$

Next, prove that 504 is the maximum value of $n$.
Let $A=\left\{a_{1}, a_{2}, \cdots, a_{n}\right\}$ be a coalition subset with the maximum number of elements.

If $a_{j}$ is the smallest number in the set $A$, it is clear that $a_{j}>1$. If $a_{j} \leqslant 1007$, then $2 a_{j} \leqslant 2014$, i.e., $2 a_{j} \in M$, and clearly, $2 a_{j} \notin A$ (because $2 a_{j}$ and $a_{j}$ have a divisibility relationship).

Consider replacing $a_{j}$ with $2 a_{j}$ in the set $A$, keeping the other elements unchanged, to form the subset $A^{\prime}$. Then $A^{\prime}$ is still a coalition subset, because for any element $a_{i}$ in $A$ other than $a_{j}$, since $a_{j}$ and $a_{i}$ are not coprime, $2 a_{j}$ and $a_{i}$ are also not coprime; and to show that $2 a_{j}$ and $a_{i}$ do not have a divisibility relationship, since $a_{j} \nmid a_{i}$, it follows that $2 a_{j} \nmid a_{i}$.

If $a_{i} \mid 2 a_{j}$, let $2 a_{j}=k a_{i}$ (clearly $k \neq 1,2$, otherwise, $a_{i}$ and $a_{j}$ would have a divisibility relationship). Then $k>2$. Thus, $a_{i} < a_{j}$, which contradicts the minimality of $a_{j}$.
Therefore, $A^{\prime}$ is still a coalition subset, and it is still an $n$-element set.
Repeat the above process until all elements in the subset are greater than 1007.
Thus, we obtain an $n$-element coalition subset
$$
B=\left\{b_{1}, b_{2}, \cdots, b_{n}\right\}\left(1007<b_{j} \leqslant 2014\right) \text {, }
$$

i.e., $B \subseteq\{1008,1009, \cdots, 2014\}$.
Since any two consecutive integers are coprime, at most 504 non-adjacent numbers can be selected from these 1007 consecutive positive integers, i.e., $n \leqslant 504$.

According to the construction mentioned earlier, the maximum value of $n$ is 504.