Air Traffic controllers (see the movie Pushing Tin). Each airline has its own callsign which gets displayed on radar screen. Each controller would be in charge of directing certain number of flights.
There are also certain air routes and altitudes that planes fly at, depending if they are heading east-ish or west-ish, they will have different 1000′ separations.
Planes now also have collision avoidance radar and procedures in case something goes wrong. Sometimes a private pilot might accidentally stray into an airline route.
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