We were enroute from Guadalajara, Mexico (MMGL) to Laredo, TX (KLRD). Laredo was a Customs stop. We only had one short leg after Laredo and we would be done for the day. It was early afternoon; the weather was beautiful.
The Big Sky, Little Airplane theory is wide open spaces, like the airspace over North central Mexico at FL370 that we were in, and ATC procedures should keep two aircraft from coming near each other, not to mention colliding. Recalling the Gol airlines flight 1907 and Embraer 600 midair collision over northern Brazil kept us from believing that theory.
While at cruise, I noticed on my MFD a TCAS surveillance target at our 12 o'clock and approximately 30 miles. The target appeared to be coming straight at us in the opposite direction and at our same altitude. We queried ATC to verify that we were supposed to be at FL370. In broken English, it was confirmed that we were correct and were assigned FL370. At this moment, I advised ATC that it appeared that we had opposite direction traffic at FL370 at approximately 25 miles. Suddenly, the controller realized his error and quickly issued a climb clearance for the opposite direction traffic while simultaneously turning us 20 degrees to the right.
TCAS clearly saved the day! Without TCAS, we would have never known that we were on a head-on collision course with the other jet. TCAS is not perfect—nothing is—but TCAS clearly dispelled the "Big Sky, Little Airplane" theory on this day.