Three. (25 points) There are 2009 athletes, numbered $1, 2, \cdots, 2009$. 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 must be selected for the honor guard at a minimum? Provide your selection plan and briefly explain your reasoning.
Three, since $45^{2}=2025>2009$, if both people's numbers are greater than or equal to 45, then their product is greater than 2009 and cannot be another person's number. If one of the two numbers is 1, then the product of 1 and the other number equals that number, which does not meet the requirement. Therefore, if the remaining people's numbers are greater than or equal to 45 or equal to 1, none of them will be equal to the product of the other two numbers.

Now take $2,3, \cdots, 44$, then the remaining people meet the requirement, so the number of people selected for the honor guard is greater than or equal to 43.

Again, take the sets $\{2,87,2 \times 87\},\{3,86,3 \times 86\}$, $\cdots,\{44,45,44 \times 45\}$ (the composition method is: $\{k, 89-k$, $k(89-k)\}$, a total of 43 groups). These written numbers are all different (because the function $y=x(89-x)$ is monotonically increasing in $(0,44.5)$), and are all between $1 \sim 2009$. If the number of people selected is less than 43, then at least one group of three numbers in the above sets will not be selected. Thus, among the remaining people, there will be someone whose number is equal to the product of the other two people's numbers.
In summary, the number of people selected should be at least 43.
(Full Tao, Nanjing University School of Engineering and Management, 210093, Hu Mingyong Nanjing Shuren International School, 210008)