7. Let $A$ be an $n$-element subset of the set $\{1,2, \cdots, 2021\}$, and any two numbers in $A$ are neither coprime nor have a divisibility relationship. Then the maximum value of $n$ is $\qquad$

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7.505 .

In the $n$-element subset of the set $\{1,2, \cdots, 2021\}$ (such that any two numbers in the subset are not coprime), the largest subset is the even subset, $\{2,4, \cdots, 2020\}$, which contains 1010 elements, and some of these elements satisfy a divisibility relationship.
Notice that, 1012 times 2 is 2024.
Thus, the set $A=\{1012,1014, \cdots, 2020\}$, where any two numbers are neither coprime nor have a divisibility relationship, contains exactly 505 elements.
In fact, 505 is the maximum value of $n$.
Because if any odd number is chosen from $\{1011,1013, \cdots, 2021\}$, it will be coprime with the adjacent even number in set $A$; if any number is chosen from $\{1,2, \cdots, 1010\}$, its multiple will be in set $A$, thus forming a divisibility relationship.