Question 111
For an integer $a \neq 0$, denote by $v_{2}(a)$ the largest non-negative integer $k$ such that $2^{k} \mid a$. For a given $n \in \mathbb{N}$, determine the largest possible cardinality of a subset $A$ of the set $\{1,2,3, \ldots, 2^{n}\}$ with the following property:
$$
\text{for all } x, y \in A, x \neq y, \text{ the number } v_{2}(x-y) \text{ is even.}
$$
$$
\text{for all } x, y \in A, x \neq y, \text{ the number } v_{2}(x-y) \text{ is even.}
$$