In order to remove warm air from a room and replacing it with a coolest air from outside, what is easier (beter performance), pumping the fresh air inside and permitting the warmer air goes out freely (blowing from outside to inside) or sucking the warm air from inside to outside (extracting) and permitting the coolest air come in freely?
Thanks and regards,
Milciades

I've done experiments with this and the answer is to blow the air out, exhaust the room, and let the cooler air be drawn in.

Most think the air blowing into the room will cool it more quick, because they feel it sooner, but the air doesn't actually cool down. They are feeling the just airflow, which is radiational cooling.

Note, this is based upon the size of the fan. If you have what's called an evacuation fan, one that move very large volumes of air, then, of course, that fan's intake need to be outside in the cooler air in order to move the cooler air in. Such a fan would still be more efficient to move the warmer air out instead.

When I put a fan on, I always have it blowing out the window. The room cools quickly, as the warm air is pushed outside and the cold air comes in through the top of the window. It seems to work better than trying to push the cold air inside.

Well, the warm air is pushed outside because the fan is inside, warm air from the room moves to replace the warm air that was pushed outside (because cold air is being pushed away from the fan). The pressure in the room is slightly lower because of the air moving, so cold air is drawn in from the top of the window.

It works best in a closed room.