The set $\{1,2,3, \ldots, 3000\}$ contains a subset of 2000 elements such that no element is double the other?
Let's construct the requested subset in the following way:

- it contains all odd numbers: $1,3,5, \ldots, 2999$. Here we already have a list with 1500 numbers.
- the set cannot contain numbers of the form $2 \times$ (odd number),
- the set can contain numbers of the form $4 \times$ (odd number), that is,

$$
\underbrace{4 \times 1}_{4}, \underbrace{4 \times 3}_{12}, \underbrace{4 \times 5}_{20}, \ldots, \underbrace{4 \times 749}_{2996}
$$

this list has 749 numbers and none of them are double the other. Moreover, none of them are double an odd number.

Thus, our set already has $1500+749=2249$ elements; therefore, any subset of it with 2000 elements meets the required conditions.