Which conflict between the united states and the soviet union nearly resulted in a third world war?

I think it was The Cuban missile crisis. Because the Soviet Union had given Cuba the missiles. JFK  then ordered the navy to form a blockade around Cuba and the disarming of the missiles.

If that isn't an option, The Soviet Union also shot down an American U-2 spy plane of Soviet air space. But it wasn't a major conflict as the pilot and a Soviet spy were traded in an agreement.