Three. (50 points) There are 2002 athletes, numbered $1, 2, 3, \cdots, 2002$. Some of them are selected to participate in the honor guard, but it must be ensured that among the remaining athletes, no one's number is equal to the product of the numbers of any two other people. How many athletes can be selected for the honor guard at a minimum? Provide your selection plan and briefly explain your reasoning.
Three, the selectable numbers for the athletes as members of the honor guard can be $2,3,4, \cdots, 44$, a total of 43 athletes, which can meet the requirements of the problem. The reason is as follows:

Because after selecting these 43 athletes as members of the honor guard, the product of any two numbers among the remaining athletes' numbers (excluding number 1) will be greater than $45^{2}=2025>2002$. And the number 1 multiplied by any number will not equal a third different number, so, among the remaining athletes, there is no one whose number equals the product of the numbers of two other athletes.

Now, we prove that 43 is the minimum number of athletes to be selected for the honor guard. That is, we prove that when 42 athletes are selected as members of the honor guard, among the remaining athletes, there are at least three numbers, where the product of two numbers equals the third number. For this, we examine the triplets:
$$
(x, 89-x, x(89-x)), x=2,3, \cdots, 43.44 \text {. }
$$

That is, $(2,87,2 \times 87),(3,86,3 \times 86),(4,85,4 \times 85)$,
$$
\cdots \cdots,(43,46,43 \times 46),(44,45,44 \times 45)
$$

a total of 43 groups.
Since the function $x(89-x)$ is monotonically increasing in the range $2 \leqslant x \leqslant 44$. Therefore, the numbers written are different and do not exceed $44 \times$ $45=1980<2002$. The number of such triplets is 43. If the number of selected athletes is less than 43, at most 42, among the above 43 triplets, at least one group of three numbers will not be selected, then these three numbers will be the numbers of the remaining athletes. One of these numbers equals the product of the other two, which does not meet the requirements of the problem. Therefore, the number of athletes to be selected for the honor guard must be at least 43.